you're allowed, the fact that you're asking us just indicates that you're afraid to just say it like it is with her, mind you, you don't seem to notice that your wife isn't afraid to say it like it is with you, ie. "not moving back home"
You don't have to be a prick or an a$$hole, just say it like it is.
The puppy isn't your pet, it's time she brought him back to her place, you already have enough animals at your place that she left behind that she doesn't take care of, it's not fair to you for her to just drop off another animal at your place, time for her to take the puppy back asap.