I don't know the laws in your state, but I sure wouldn't do anything illegal about it. Far too many horrible things could happen. Think about potential consequences before doing anything rash. The wrong decision could not only haunt you for a long time to come, it could also reflect badly on all bowhunters. We sure don't need bad PR about some dog getting shot. And believe me, the media would portray the hunter as the villian regardless of the explanation.
Deal with the owner if possible, or the game warden or law enforcement official who handles that type of problem.
Trust me, loosing an arrow might seem tempting and a quick solution to your problem. But it could very quickly turn into just the beginning of a much much larger problem.