Rex is a Rotweiler that appears in the episode "Dances with Dogs".
Bill buys a dog so he can enter the dog-dancing competition himself after Bobby and Hank enter the contest (separately). Rex was extremely aggressive and hostile towards Bill, attacking him anytime he removes his muzzle. After he trapped Rex in the shower, Bill subdues the dog with sleeping pills and takes him to the competition, but the judge has a no muzzles rule. Bill complies and Rex continues to attack him the second his muzzle is removed. It was not mentioned what happens to him afterwards, but it can be assumed that Bill had returned him to the animal shelter.
It's notable that Bill's next experience with dogs in The Petriot Act was in complete contrast to his time with Rex. In that episode Bill provided a temporary foster home for Buster, who was a talented and obedient Golden Retriver and Hank was the one with an uncontrollable pet.