I don't think any current soulbind would change Cait in such a small way that it would actually be barely any work. At least, none of the soulbinds we currently see in-game. Unless you're thinking of whatever that merc leader has, which I think can only be done to someone before they're born, because that doesn't seem to have a cosmetic change associated. But amongst the three options the Champ will have, you're either severely under-estimating the difference between a fox and a cat, or you mean the valkyrie option (which, given Cait's appearance, might give pretty much no change to her besides bolting wings onto her back), because I'm, certain you don't think turning her into a lizard is a small change.I think you're both missing how little a change I'd be okay with - and that this a thread asking what people would like to see.
Aside from all this, I doubt any companion can get a soulbind, even if we could wave a wand and say all the work has been done to make scenes. In-story, I mean. The gods really want your soul for the same reasons as Kas: your soul is a really juicy one. Massive amounts of power, and tons of potential. Because of it, the Champ is extremely malleable, and also, desirable for any soul-suckers.