Look, you got Cart Blanche from the Officer who came by. You have an agressive dog in your yard that is a threat to your family. Don't worry about causing a scene with the neighbors! First off, most people won't approach you when you are armed. Second, you should probably talk to your neighbors and explain what's going to go down-you might find them willing to chip in for ammo.
Oh, I wouldn't worry too much about legal action from the owners. First off, it sounds as though they are already conversant with the Justice System and probably don't want anything to do with it. Second, anything more than small claims court will cost them money. Doesn't sound like they have any. Even if they did, the officer could be subpoena'd and it would be Game Over for them.
Dave
ETA: The dog is an aggressive asshole. It's a crying shame he was raised that way, but he is what he is. I wouldn't feel any worse about capping his ass than if the thrashy neighbors raised a thug kid and he was in your yard threatening your family with bodily harm.