Fitting model to intensity plot

I am trying to fit a model to the following dataset to extract numerical values for 4 parameters $ J_x$ , $ J_y$ , $ J_z$ , and $ g$ . I also know that $ g \approx 2$ in this case. The dataset is a list of quadruples: {x, energy, intensity, error} as shown below

dataset = {{0.01299648, 0.01203211, 0.1263361, 0.005950636}, {0.01299648, 0.04910681, 0.0336076, 0.002947696}, {0.01299648, 0.09977061, 0.001322289, 0.000413821}, {0.01299648, 0.1508783, 0.000499663, 0.000258259}, {0.01299648, 0.2008796, 0.000419055, 0.00024877}, {0.01299648, 0.2510364, 0.000421737, 0.000272571}, {0.01299648, 0.3009251, 0.000178943, 0.000156955}, {0.01299648, 0.3508747, 0.0000992, 0.0000883}, {0.01299648, 0.3999321, 0.000430162, 0.000468312}, {0.01299648, 0.4489179, 0.001252234, 0.000846992}, {0.01299648, 0.5002585, 0.000617269, 0.000553035}, {0.01299648, 0.5509165, 0.001468457, 0.000842178}, {0.01299648, 0.6011173, 0.003723728, 0.001349723}, {0.01299648, 0.6498302, 0.004062989, 0.001265983}, {0.01299648, 0.6988636, 0.001993023, 0.000906512}, {0.01299648, 0.7499531, 0.000721637, 0.000587884}, {0.01299648, 0.8010127, 0.000252952, 0.000316284}, {0.05334629, 0.01203211, 0.1305249, 0.004184997}, {0.05334629, 0.04910681, 0.03503799, 0.002187056}, {0.05334629, 0.09977061, 0.001494748, 0.000322744}, {0.05334629, 0.1508783, 0.000631434, 0.000216124}, {0.05334629, 0.2008796, 0.000516482, 0.000212526}, {0.05334629, 0.2510364, 0.000452927, 0.000203133}, {0.05334629, 0.3009251, 0.00038714, 0.000173926}, {0.05334629, 0.3508747, 0.000419254, 0.000179236}, {0.05334629, 0.3999321, 0.000425151, 0.000310161}, {0.05334629, 0.4489179, 0.000511058, 0.000412408}, {0.05334629, 0.5002585, 0.000683154, 0.000400352}, {0.05334629, 0.5509165, 0.001937698, 0.000617178}, {0.05334629, 0.6011173, 0.003902016, 0.000892543}, {0.05334629, 0.6498302, 0.00309874, 0.000839511}, {0.05334629, 0.6988636, 0.001156821, 0.000561058}, {0.05334629, 0.7499531, 0.000876003, 0.000445513}, {0.05334629, 0.8010127, 0.000494271, 0.000353135}, {0.05334629, 0.8507249, 0.000468474, 0.000405826}, {0.05334629, 0.9009042, 0.000227273, 0.000284132}, {0.09946869, 0.01203211, 0.1314684, 0.002607759}, {0.09946869, 0.04910681, 0.03540794, 0.001393436}, {0.09946869, 0.09977061, 0.001480958, 0.000215194}, {0.09946869, 0.1508783, 0.000518701, 0.000144396}, {0.09946869, 0.2008796, 0.00039427, 0.000139293}, {0.09946869, 0.2510364, 0.000395253, 0.000148412}, {0.09946869, 0.3009251, 0.000357242, 0.000144663}, {0.09946869, 0.3508747, 0.000435539, 0.000173756}, {0.09946869, 0.3999321, 0.000440639, 0.000186949}, {0.09946869, 0.4489179, 0.000453975, 0.000189049}, {0.09946869, 0.5002585, 0.000817151, 0.000265822}, {0.09946869, 0.5509165, 0.002821699, 0.000522746}, {0.09946869, 0.6011173, 0.005799377, 0.000790837}, {0.09946869, 0.6498302, 0.003142505, 0.000619193}, {0.09946869, 0.6988636, 0.00089875, 0.000377157}, {0.09946869, 0.7499531, 0.000935933, 0.000440654}, {0.09946869, 0.8010127, 0.000741281, 0.00040915}, {0.09946869, 0.8507249, 0.000379727, 0.000311241}, {0.09946869, 0.9009042, 0.000452129, 0.000372186}, {0.09946869, 0.9499643, 0.000321497, 0.000298012}, {0.09946869, 0.9987899, 0.000216047, 0.000265462}, {0.09946869, 1.049146, 0.000321083, 0.000405456}, {0.09946869, 1.100007, 0.00007, 0.000197968}, {0.09946869, 1.151003, 0.000371132, 0.000898138}, {0.09946869, 1.20088, 0.001603127, 0.001909191}, {0.1513923, 0.01203211, 0.1271927, 0.002382964}, {0.1513923, 0.04910681, 0.03466543, 0.001207942}, {0.1513923, 0.09977061, 0.001471831, 0.000180972}, {0.1513923, 0.1508783, 0.000458085, 0.000108564}, {0.1513923, 0.2008796, 0.000323088, 0.0000968}, {0.1513923, 0.2510364, 0.000262988, 0.0000927}, {0.1513923, 0.3009251, 0.000208331, 0.0000952}, {0.1513923, 0.3508747, 0.000261884, 0.000132211}, {0.1513923, 0.3999321, 0.000318829, 0.000154663}, {0.1513923, 0.4489179, 0.000390393, 0.000170512}, {0.1513923, 0.5002585, 0.000796483, 0.000230864}, {0.1513923, 0.5509165, 0.003169181, 0.00045884}, {0.1513923, 0.6011173, 0.006016299, 0.000634249}, {0.1513923, 0.6498302, 0.002961305, 0.000479318}, {0.1513923, 0.6988636, 0.00089255, 0.00030299}, {0.1513923, 0.7499531, 0.000601942, 0.000292041}, {0.1513923, 0.8010127, 0.000622036, 0.000308231}, {0.1513923, 0.8507249, 0.00059582, 0.000311122}, {0.1513923, 0.9009042, 0.000342461, 0.000258311}, {0.1513923, 0.9499643, 0.000365842, 0.000264302}, {0.1513923, 0.9987899, 0.000383168, 0.000282596}, {0.1513923, 1.049146, 0.000158197, 0.000218464}, {0.1513923, 1.100007, 0.0000797, 0.0001487}, {0.1513923, 1.151003, 0.000272186, 0.000602807}, {0.1513923, 1.20088, 0.000791483, 0.000943182}, {0.1513923, 1.24981, 0.000810134, 0.000846855}, {0.1513923, 1.298876, 0.001098106, 0.000878741}, {0.1513923, 1.35012, 0.001020097, 0.00091243}, {0.1513923, 1.400843, 0.00099628, 0.001472998}, {0.1513923, 1.45113, 0.001696679, 0.002615917}, {0.1513923, 1.487999, 0.00068497, 0.001417412}, {0.200413, 0.01203211, 0.1251366, 0.002421743}, {0.200413, 0.04910681, 0.03396657, 0.001239589}, {0.200413, 0.09977061, 0.001455514, 0.000186787}, {0.200413, 0.1508783, 0.000471295, 0.000107087}, {0.200413, 0.2008796, 0.000356116, 0.0000954}, {0.200413, 0.2510364, 0.000255785, 0.0000834}, {0.200413, 0.3009251, 0.000192659, 0.0000751}, {0.200413, 0.3508747, 0.000195415, 0.000086}, {0.200413, 0.3999321, 0.000212672, 0.0000972}, {0.200413, 0.4489179, 0.000201986, 0.0000998}, {0.200413, 0.5002585, 0.000452148, 0.00014457}, {0.200413, 0.5509165, 0.002283648, 0.000302595}, {0.200413, 0.6011173, 0.005130702, 0.000435712}, {0.200413, 0.6498302, 0.002947664, 0.000341739}, {0.200413, 0.6988636, 0.000885883, 0.00019889}, {0.200413, 0.7499531, 0.000353508, 0.000142375}, {0.200413, 0.8010127, 0.000337989, 0.000159135}, {0.200413, 0.8507249, 0.000294789, 0.00016511}, {0.200413, 0.9009042, 0.000287179, 0.00017037}, {0.200413, 0.9499643, 0.000311635, 0.000194681}, {0.200413, 0.9987899, 0.000207756, 0.000158586}, {0.200413, 1.049146, 0.000158257, 0.00013942}, {0.200413, 1.100007, 0.000190184, 0.000146221}, {0.200413, 1.151003, 0.000213257, 0.000208235}, {0.200413, 1.20088, 0.000336925, 0.000306631}, {0.200413, 1.24981, 0.000487695, 0.000424801}, {0.200413, 1.298876, 0.000638927, 0.000549385}, {0.200413, 1.35012, 0.001054225, 0.000776587}, {0.200413, 1.400843, 0.001720866, 0.001358017}, {0.200413, 1.45113, 0.00200075, 0.001706196}, {0.200413, 1.487999, 0.001241234, 0.001336194}, {0.249747, 0.01203211, 0.1205826, 0.002361229}, {0.249747, 0.04910681, 0.03260196, 0.001191631}, {0.249747, 0.09977061, 0.001261705, 0.000170178}, {0.249747, 0.1508783, 0.000333223, 0.00009}, {0.249747, 0.2008796, 0.000305626, 0.0000888}, {0.249747, 0.2510364, 0.000248531, 0.0000826}, {0.249747, 0.3009251, 0.000227554, 0.0000828}, {0.249747, 0.3508747, 0.000242098, 0.0000856}, {0.249747, 0.3999321, 0.000213059, 0.0000796}, {0.249747, 0.4489179, 0.000171657, 0.0000691}, {0.249747, 0.5002585, 0.000269242, 0.0000875}, {0.249747, 0.5509165, 0.001302021, 0.000202058}, {0.249747, 0.6011173, 0.003488679, 0.000339568}, {0.249747, 0.6498302, 0.003051286, 0.000322039}, {0.249747, 0.6988636, 0.001461859, 0.000227405}, {0.249747, 0.7499531, 0.000595035, 0.000144464}, {0.249747, 0.8010127, 0.000388456, 0.000118404}, {0.249747, 0.8507249, 0.000291368, 0.00010498}, {0.249747, 0.9009042, 0.000241225, 0.000107813}, {0.249747, 0.9499643, 0.000237664, 0.00011481}, {0.249747, 0.9987899, 0.000151976, 0.0000917}, {0.249747, 1.049146, 0.000151201, 0.000093}, {0.249747, 1.100007, 0.000175435, 0.000106715}, {0.249747, 1.151003, 0.000146402, 0.00010687}, {0.249747, 1.20088, 0.000207344, 0.0001413}, {0.249747, 1.24981, 0.000247186, 0.00017871}, {0.249747, 1.298876, 0.000406102, 0.000266496}, {0.249747, 1.35012, 0.000704678, 0.00045794}, {0.249747, 1.400843, 0.000829369, 0.000665319}, {0.249747, 1.45113, 0.000937308, 0.000875315}, {0.249747, 1.487999, 0.001484896, 0.00116244}, {0.2999469, 0.01203211, 0.1134484, 0.002063204}, {0.2999469, 0.04910681, 0.03021775, 0.001057232}, {0.2999469, 0.09977061, 0.00106177, 0.000140285}, {0.2999469, 0.1508783, 0.00030537, 0.0000768}, {0.2999469, 0.2008796, 0.000213005, 0.0000649}, {0.2999469, 0.2510364, 0.000185747, 0.0000657}, {0.2999469, 0.3009251, 0.000164455, 0.000065}, {0.2999469, 0.3508747, 0.000142963, 0.0000636}, {0.2999469, 0.3999321, 0.000130967, 0.0000592}, {0.2999469, 0.4489179, 0.000167198, 0.0000687}, {0.2999469, 0.5002585, 0.000180923, 0.0000726}, {0.2999469, 0.5509165, 0.000541689, 0.000134283}, {0.2999469, 0.6011173, 0.001467756, 0.000226035}, {0.2999469, 0.6498302, 0.00203471, 0.000272955}, {0.2999469, 0.6988636, 0.001823599, 0.000254749}, {0.2999469, 0.7499531, 0.001339045, 0.000208357}, {0.2999469, 0.8010127, 0.0009652, 0.000182255}, {0.2999469, 0.8507249, 0.00046882, 0.000122285}, {0.2999469, 0.9009042, 0.000292738, 0.000102422}, {0.2999469, 0.9499643, 0.000180685, 0.0000821}, {0.2999469, 0.9987899, 0.000126662, 0.0000694}, {0.2999469, 1.049146, 0.000182039, 0.000087}, {0.2999469, 1.100007, 0.000243545, 0.0000985}, {0.2999469, 1.151003, 0.000167506, 0.0000907}, {0.2999469, 1.20088, 0.000221046, 0.000117207}, {0.2999469, 1.24981, 0.000266948, 0.000130003}, {0.2999469, 1.298876, 0.000361939, 0.000171356}, {0.2999469, 1.35012, 0.000423017, 0.000222659}, {0.2999469, 1.400843, 0.000497565, 0.000299089}, {0.2999469, 1.45113, 0.000841964, 0.00069896}, {0.2999469, 1.487999, 0.001636413, 0.001292695}, {0.351399, 0.01203211, 0.1023983, 0.001765024}, {0.351399, 0.04910681, 0.02760775, 0.000890958}, {0.351399, 0.09977061, 0.001013449, 0.000120122}, {0.351399, 0.1508783, 0.000314347, 0.0000681}, {0.351399, 0.2008796, 0.000218643, 0.0000587}, {0.351399, 0.2510364, 0.000145264, 0.0000501}, {0.351399, 0.3009251, 0.0000988, 0.0000431}, {0.351399, 0.3508747, 0.0000886, 0.0000424}, {0.351399, 0.3999321, 0.000078, 0.0000407}, {0.351399, 0.4489179, 0.000113312, 0.0000492}, {0.351399, 0.5002585, 0.0000947, 0.0000463}, {0.351399, 0.5509165, 0.000150304, 0.0000649}, {0.351399, 0.6011173, 0.000443105, 0.000115168}, {0.351399, 0.6498302, 0.000857112, 0.000160557}, {0.351399, 0.6988636, 0.001158728, 0.000179657}, {0.351399, 0.7499531, 0.001562672, 0.000200422}, {0.351399, 0.8010127, 0.001445169, 0.000188441}, {0.351399, 0.8507249, 0.000932427, 0.0001463}, {0.351399, 0.9009042, 0.00044006, 0.00010092}, {0.351399, 0.9499643, 0.000182093, 0.00007}, {0.351399, 0.9987899, 0.000100358, 0.0000596}, {0.351399, 1.049146, 0.000133863, 0.0000725}, {0.351399, 1.100007, 0.000144129, 0.000072}, {0.351399, 1.151003, 0.00010871, 0.0000655}, {0.351399, 1.20088, 0.000123393, 0.0000805}, {0.351399, 1.24981, 0.000193871, 0.000102966}, {0.351399, 1.298876, 0.000241517, 0.000134903}, {0.351399, 1.35012, 0.000296181, 0.000181252}, {0.351399, 1.400843, 0.00026523, 0.000220609}, {0.351399, 1.45113, 0.000646058, 0.000572446}, {0.351399, 1.487999, 0.000963321, 0.000939469}, {0.4027665, 0.01203211, 0.09461402, 0.001366129}, {0.4027665, 0.04910681, 0.0254652, 0.000706637}, {0.4027665, 0.09977061, 0.001031896, 0.0000992}, {0.4027665, 0.1508783, 0.000384627, 0.0000604}, {0.4027665, 0.2008796, 0.000268799, 0.0000534}, {0.4027665, 0.2510364, 0.000208381, 0.0000464}, {0.4027665, 0.3009251, 0.000199062, 0.0000445}, {0.4027665, 0.3508747, 0.000186287, 0.0000418}, {0.4027665, 0.3999321, 0.000159836, 0.0000416}, {0.4027665, 0.4489179, 0.000107729, 0.000037}, {0.4027665, 0.5002585, 0.0000923, 0.0000347}, {0.4027665, 0.5509165, 0.0000702, 0.0000317}, {0.4027665, 0.6011173, 0.000146609, 0.0000496}, {0.4027665, 0.6498302, 0.000280341, 0.0000714}, {0.4027665, 0.6988636, 0.000459356, 0.0000931}, {0.4027665, 0.7499531, 0.000744988, 0.00011817}, {0.4027665, 0.8010127, 0.001140008, 0.000143582}, {0.4027665, 0.8507249, 0.001200483, 0.000141774}, {0.4027665, 0.9009042, 0.000922343, 0.000119712}, {0.4027665, 0.9499643, 0.000375711, 0.0000757}, {0.4027665, 0.9987899, 0.000140653, 0.0000492}, {0.4027665, 1.049146, 0.000098, 0.0000461}, {0.4027665, 1.100007, 0.0000772, 0.0000404}, {0.4027665, 1.151003, 0.0000815, 0.0000407}, {0.4027665, 1.20088, 0.0000899, 0.0000479}, {0.4027665, 1.24981, 0.0000992, 0.0000589}, {0.4027665, 1.298876, 0.000142484, 0.0000848}, {0.4027665, 1.35012, 0.000189181, 0.000109538}, {0.4027665, 1.400843, 0.000172183, 0.000138258}, {0.4027665, 1.45113, 0.000275446, 0.000293852}, {0.4027665, 1.487999, 0.000381214, 0.000439755}, {0.4496463, 0.01203211, 0.1094655, 0.001215262}, {0.4496463, 0.04910681, 0.03120829, 0.000643984}, {0.4496463, 0.09977061, 0.001421187, 0.0000962}, {0.4496463, 0.1508783, 0.000589879, 0.0000633}, {0.4496463, 0.2008796, 0.000401797, 0.0000524}, {0.4496463, 0.2510364, 0.000392448, 0.0000514}, {0.4496463, 0.3009251, 0.000374653, 0.0000495}, {0.4496463, 0.3508747, 0.000443615, 0.0000546}, {0.4496463, 0.3999321, 0.000371195, 0.0000521}, {0.4496463, 0.4489179, 0.000203434, 0.0000406}, {0.4496463, 0.5002585, 0.000117548, 0.0000317}, {0.4496463, 0.5509165, 0.0000764, 0.000026}, {0.4496463, 0.6011173, 0.0000956, 0.0000299}, {0.4496463, 0.6498302, 0.000120406, 0.0000352}, {0.4496463, 0.6988636, 0.000175619, 0.000043}, {0.4496463, 0.7499531, 0.000325115, 0.0000604}, {0.4496463, 0.8010127, 0.0005661, 0.0000836}, {0.4496463, 0.8507249, 0.000928296, 0.000105136}, {0.4496463, 0.9009042, 0.001128707, 0.000111962}, {0.4496463, 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0.000248875, 0.000441169}} 

My model yields the energy at a specific coordinate $ \boldsymbol{k} =\{k_x, k_y, k_z\}$ where $ \boldsymbol{k} = x(\boldsymbol{b}_1 + \boldsymbol{b}_2)$ . Here $ x$ is the first entry of each quadruple in the dataset, $ \boldsymbol{b}_1 = \{ 2 \pi, 2\pi/\sqrt{3}, 0 \}$ , and $ \boldsymbol{b}_2 = \{ 0, 4\pi/\sqrt{3}, 0 \}$

(* First obtain the measured energy values and store them in a list *) ω = DeleteDuplicates@*Flatten@dataset[[All, 2]]; b1 = {2π, (2π)/Sqrt[3], 0}; b2 = {0, (4π)/Sqrt[3], 0}; xKtoM = DeleteDuplicates@*Flatten@dataset[[All, 1]]; k = Table[xKtoM[[i]] (b1 + b2), {i, 1, Length[xKtoM]}]; 

I then go on to define my model

(* Bohr magneton μ in units of eV*T^-1 pulled from wikipedia*) μ = 5.7883818012*10^-5;  (* This is the model where we have taken the positive branch of the energy spectrum *) d[kx_, ky_, kz_, g_, Bz_, Jx_, Jy_, Jz_] := {1/2 (Jx + Jy) (Cos[k[[1]]] + Cos[k[[2]]] + Cos[k[[1]] + k[[2]]]) - 3 Jz + g \[Mu] Bz,  1/2 (Jx - Jy) (E^(-I 2 \[Pi]/3) Cos[k[[1]]] + E^(I 2 \[Pi]/3) Cos[k[[2]]] + Cos[k[[1]] + k[[2]]])};  spectrum[{kx_, ky_, kz_}] := Norm[d[kx, ky, kz, g, 4, Jx, Jy, Jz]]; (* Get the spectrum going from the high symmetry K \[Rule] M path using the k vector/list we defined above. This is the model we use for the dataset *) model = Map[spectrum, k]; 

From here I would have the energy values at 18 separate points if I had specified values for Jx, Jy, Jz, and g. I then thought to define a cost function that takes my model and subtracts the corresponding measured energy value and then try to find the minimum using the built-in FindMinimum function

(* Define a cost function between the model and the measured energy values Subscript[\[Omega], i] *) CostFunction[Jx_, Jy_, Jz_, g_] := Sum[Abs[model[[i]] - ω[[i]]]^2, {i, 1, Length[ω]}] (* Minimize this cost function to attempt to extract the parameters Jx, Jy, Jz, and g *) params = FindMinimum[model, {{Jx, 0.5}, {Jy, 0.5}, {Jz, 0.5}, {g, 2}}] 

My only issue with this is that

  1. My model is a list of functions while the measured energy values $ \omega$ is a list of real numbers
  2. Using FindMinimum yields quite a lot of errors

Is it possible to get around these issues using something like NonLinearModelFit or something similar? Let me know if I should clarify anything or add any additional info and thanks in advance!