Not really gettin' the question- my liability insurance covers any bad thing that happens to folks as a result of my actions, which includes anything my dog/other pets might do. This unless the animal's type/breed is specifically restricted in the policy. Homeowner's is first line of defense, but I actually had a Farmer's auto liability policy pay off one time in a dog bite incident, and it happened when the dog got off her tieout at home! Nothin' whatever to do with the car....
Unless this particular animal has already been involved in some sort of kerfuffle, I don't understand the need for a stand alone policy-?