Could you allow players to move pets from their stall onto an alt on the same account, and back? I think allowing the pet to be unable to breed after transfer and the pet unusable by the 2nd character would be worth it.
Reason being, I enjoy the pets I have and I would be interested in breeding different types as they come out, but I don't think I have the animal slots to do this while actively using animal handling as a combat skill. I have some of the better bees, and obviously I have he option to release them, but I really don't like that idea. The players into breeding have characters dedicated to breeding as that's the easiest way to manage the slot count.
I think a better balance of maintaining the difficulty of animal husbandry while letting players have flexibility in what pets they have available for combat could be met.