I suggest a new noble position, Chief Veterinarian, the animal equivalent of the Chief Medical Dwarf.
Injured pets could be taken to the veterinary clinic (a hospital zone for animals) by their owners. Injured stray animals could be taken by any dwarf with the Animal Care labor turned on. In addition, animal caretakers will bring food/water to hungry/thirsty animals (I don't think most animals drink, but stray tigermen do). The clinic would have cages or chains instead of beds, and the animals would be restrained until they were discharged.
Injured animals in the veterinary clinic are treated just like injured dwarves in the hospital, using the same medical skills.
This will give players an option other than euthanasia when dealing with injured animals, allow medical dwarves to practice their medical skills during un!!FUN!! times without committing atrocities against migrants (not that there's anything wrong with that), and makes the Animal Care labor useful.
You could, at least initially, forgo the new noble and zone, and just let the CMD and hospital perform double-duty, but some (perhaps many?) players might like to keep them separated, if only to ensure that limited medical supplies are used on dwarves first.
Sorry if this has been suggested before. I searched, but nothing similar came up.