I tried to optimize it as much as possible and only do the re-baking when there is actually an animation going on, but during animations that go on all the time when the item is held, the memory usage increases at least 5 times faster than normal and the frame rate becomes less stable...

if the animation is repetitive, did you use a cache where you store the baked models? you should access it when there's an animation to see if there's already a baked model with the data you want and if not re-bake and add to the cache. Prevents re-baking on every animation tick

I have a float that describes the animation progress (varies between 0 and 1) and my in my animastions models have to move around according to that value. I can't save every single step of the animation because there are about a million of possible states...

can't you at least make the number of values that that float can be as least as possible? I mean sure you won't have to use all these million possible values, maybe rounding to a precision or something.

The thing is that there are so many more factors that influence the model like paintjobs and attachments to the item.