Can you get visitation rights for pets? (like you do with kids)?

Depends on who bought the dog. If you bought it, your name is on the tags and registered in your name, its your property (because animals are viewed as property) so he would have to give the dog to you. If he bought it, you're out of luck. Try buying the dog from him (if you do, make sure you get proof, so he can't come back later and claim you stole it), or getting him to agree to setting up a way so you can see the dog w/o seeing him (like have a friend bring the dog by)