Not sure why you'd bother. Bards are full casters and Arcane Tricksters are 1/3 casters, so the Bard is going to be better, caster-wise.
If the goal is to get the Mage Hand tricks for Bard, I'd probably disallowing it on the premise that that's one of the Rogue schticks. Class/level based systems have a myriad of legitimate issues, but they have benefits, too. There's a reason why I keep coming back to D&D after 30+ years of gaming. If you're doing a class/level system, you have to respect the boundaries or things go south. Leave the walls intact. YMMV, but that's my experience.
If I were the DM, I'd probably encourage the player to play a Rogue to 3rd level, then switch to Bard (or Bard 1, Rogue 3, back to Bard, depending on cherries to pick).
That said, it's your game. You may have different play goals than I do. I don't see anything concretely "wrong" or "unbalanced" with the idea. I just "smells wrong" and I suspect that you'll end up with some unintended consequences. I might be wrong. Even if I'm right, it can be a lot of fun to see how tweaks like that impact the system first-hand.
Short form: I don't advise it, but I'm not your mother and it's not bad-wrong-fun.