fearsomepirate
Hero
Oh, ok. I thought you were doing a dual-wielding Rogue that had dipped into Bladesinger for whatever reason. Nevermind, then. It works exactly how you'd expect in the OP.
You need to be a 6th level wizard to get multiattack. I am not there yet, I am currently 3-2Skimming over the stuff, I'd probably allow it. I think the limiting factor to its benefit is you can only only do it in light or no armor, use two hands on a weapon, it ends if you're incapactitated, and you can also only do it a number of times equal to your proficiency bonus per long rest.
It's not the horror story of the Champions character maxed out to only do the Move Through action every round for damage.
EDIT: I'm curious. What level are you at to be an arcane trickster/bladesinger? You'd have to be at least a level 3 thief/level 3 wizard?