I think it’s generally a culture thing; you don’t eat your beasts of burden (oxen, horses, etc) and you don’t eat your animaux de compagnie. Not just because they’re useful and you would more than likely have a connection to them, but generally speaking, these are very lean animals, barely any fat on them, plus the fact that unless you know what you’re doing and make the slaughtering of your animal quick and painless, endocrines get released into the muscle tissue, making the meat bitter.
Spearmint would be good too if you’re going for a slightly sweeter, less intense flavor than peppermint!