Foam clay is a type of moldable EVA foam that can be cut, sanded, and drilled after it dries. It is very lightweight and flexible and is used in costuming and props a lot, which is why I thought it might be a good fit for an animated prop so it wouldn't cause too much strain on the motor used. But it is water-based, so I was afraid it might not hold up well. It still might, if it's coated in something protective like Plasti-Dip before painting.
But all that to say, I will look into foam wood epoxy! Even if it isn't a good fit for this project, there are always others.