Currently, the list of non-movable block entities is hard-coded, as shown below:

Would it be possible to take that list, put it in a tag, and make movableBlockEntities use that tag when determining whether or not to let that block entity move?

maybe a Predicate (https://minecraft.wiki/w/Predicate) is better. you can write blockstates inside it, but block tag can only have block ids.
i m thinking using a special predicate __scarpet:piston_movable is considerable.
i use this name is because that __scarpet_marker is a special name already used by this mod.(here)