I managed to fix the problem, thanks to everyone who took the time to provide help/advice. I managed to open the motor but it is impossible to do that without damaging the clips that are holding it in place.
As for the cause of the problem itself: the lubricant that was used on the tiny white planetary gears inside wasn't distributed equally throughout the orange bearings in the carrier, which created this bump that caused the whole mechanism to shift upward whenever the motor started spinning. I removed all planetary gears and applied the excessive lubricant to all of them and reassembled the motor - which fixed the wobble, but it also fixed that nasty sound it was making.
Hopefully, this will help anyone who is suffering from the same problem, but beware: you will damage the clips. But it shouldn't be that bad.
