Perhaps change the name from "pet" to "familiar" for those that have issues with it.
Also, having a pet that can be upgraded different ways, and is always active in play and can't be targeted, like a Mark, would be a decent idea if they weren't made completely overpowered. Also, they wouldn't be able to deal damage each turn like weapons, only if their ability allowed it, for a cost. Or, better yet, have a variety of Familiars that can just be changed out, much like a Mark, and each one has a different ability for different elements. One, perhaps, can deal one damage to target creature at a certain cost, another could perhaps heal a creature for one HP, and so on.