Sounds reasonable to me. One thing that stood out to me is the Animal Empathy advantage; it would allow normal interaction skills to be used on animals anyway. But the Animal Handling skill as you mention it seems better; not all animal trainers should need to be diplomatic, persuasive people persons, after all. Some at least probably prefer the company of animals.
Would you allow other little pets, as Featurs, or something? Like, a songbird that tended to follow the character around, or a hatchling riding-bird he kept in a saddlebag, or whatnot. Or anything else, really, since the description of Musians says they train other animals as well.