Maybe there could be an illegal animal pen somewhere outside Riften or far out somewhere in the mountains of The Reach where you could buy animals by some shady dude in a hood

Possible animals that I think would work and not be "to much" is sabre cats, wolfs and bears. And they would not be cheap, maybe 3000-10000 gold per animal.
This would not be to hard for Bethesda to fix in a future patch, right?