Is Booming Blade that unbalanced? How could it be rebalanced?

Up until now, we had quite the terrible party lineup (a Rogue, two Paladins and a Barbarian; we soon discovered the game is simply not enjoyable without casters), and we all decided to fix this by designing a new party together: our next lineup will consist of the same Rogue and the Barbarian, but the Paladins (one of which was me) will be replaced by a Divination Wizard and an unholyly multiclassed Bard. Since the Wizard and the Rogue are quite squishy, we decided I should take the role of a frontliner, so I designed the character to be at least kind of an off-tank.

Long story short, I built my character based around a sword and board build with the Ruby of the War Mage and the War Caster feat. Flavour-wise, this character was designed to be both air-headed and hot-headed (hot-air-balloon-headed, if you will), and as such is prone to engage in melee combat if only to feel the thrill of the fight. Since I will only pick three levels of Warlock (the Pact of the Tome is integral to the flavour of my character concept), I won’t be able to access the Pact of the Blade’s Extra Attack invocation. Thus, I based the character’s melee DPR output around Booming Blade and Green Flame Blade.

I discussed with my GM the possibility of being able to trigger Booming Blade via Command, and he said it was not possible, which is reasonable. However, he started looking into the cantrip itself and some (hastily) made damage comparison tables, and said it is broken and should be nerfed to avoid overshadowing the martial characters. Some of the proposed nerfs were:

  • Blade cantrips are now a level 1 spell.
  • Blade cantrips don’t trigger the primary effect, only the secondary one (that is, BB’s movement damage and GFB’s "cleaving" damage)
  • Blade cantrips don’t trigger the primary effect, but the secondary one is increased by one die size (BB’s movement and GFB’s "cleaving" would use d12)
  • Reduced dice size for the cantrip’s damage (for example, d8 to d6).
  • Reduced dice size for the cantrip’s primary effect, but augmented for the secondary effect (d8 to d12)

Needless to say, I’m kinda bummed by this change since the whole gish appeal of this character concept was based around being able to use these cantrips, since playing a melee character with a single attack, only 1d6 at Sneak Attack and the very limited magic items we have would suck hard; I would feel stupid using a single attack when I could be spamming Eldritch Blast instead. The secondary effects of these cantrips are extremely situational and I am sure they won’t come up many times since we often face intelligent enemies. I would be kinda okay with the last proposal, but still, I’m not convinced. I proposed the DM to write some homebrew (I consider myself fairly good at balancing) martial weapons and mechanics to make martials more flexible instead of outright nerfing the cantrip, but he said he wants this campaign to be as vanilla as possible.

How can I convince my DM that the Blade cantrips aren’t that unbalanced? Alternatively, what’s a reasonable way to rebalance them?