Let's flip this base a little more! Maybe it'll shake some dust off.
The blueprints for the base were hand-delivered by Dr.Schultz...
In other words, the blueprints themselves could have been tampered with to allow the cube to fall down into the base without getting stuck in the sides.
Since she also supervised the installation of the climate control mechanisms, there could have been enough of a gap left to swallow the cube. And there could be a remote mechanism to flip the base, one that would make a whirr sound when activated and a tink sound once the cube falls down. As long as the card is placed on the other side of the flipped base, the trick easily works.
But what about not damaging the engine?
Well, it's simple: we know the object inside the case was the true engine, but that doesn't mean
of it is authentic. I suggest that the culprit 3d-printed an identical looking protective shell to encase the true engine with. The difference in size would be small enough that nobody would notice when staring through the case, and that way, the protective layer would protect the engine from incurring any damage from the fall.