Numerical issues in Fourier transform of Mathieu functions

I’m trying to calculate the Fourier transform of the Mathieu $ \text{me}$ functions using NIntegrate but keep on getting NIntegrate::inumr (the integrand has evaluated to non-numerical values) and NIntegrate:ncvb (failed to converge to prescribed accuracy) errors. Here is how I’m defining the Mathieu functions

ce[r_, z_, q_] := MathieuC[MathieuCharacteristicA[r, q], q, z] se[r_, z_, q_] := MathieuS[MathieuCharacteristicB[r, q], q, z] me[r_, z_, q_] := (ce[r, z, q] + I se[r, z, q]) / Sqrt[2 Pi] 

and I’m evaluating the integral

NIntegrate[Exp[-I n z] / Sqrt[2 Pi] me[r, z, q], {z, 0, 2 Pi}] 

where r is very close to an integer (I avoid exact integers because MathieuCharaceristicA and MathieuCharaceteristicB are not continuous at integers) and q for example varies continuously from -70 to 0.

Does anyone know how I can get around the numerical issues? Or is there a more efficient way to numerically Fourier transform Mathieu functions?