Oh, I am so sorry about Billy. I can empathize. I had two easter egger sisters, and one, Bonita, got killed in my backyard...the neighbor let out the two savage chicken-eating dogs onto my lawn (for the umpteenth time). I was so furious when he came out of the house holding my bird by the ankles (his dog carried it home). (I also made him clean up his dogs poops right in front of me, I was so angry)
When I got over my initial fury and tears, I wrote a letter explaining how much a bird like Bonita means to me. I explained that she is a treasured pet and that she is also worth $ (not much, but some). I explained to him that he must start taking responsibility for his dogs actions (as he was neglecting to all around).
In order for us to remain at all civil, I asked him to present me with $30 for Bonita (for her value as an egg layer and to pay for a replacement hen for her sister). I think 30$ stung just enough to make him pay attention. Furthermore, I copied the local laws and highlighted how dogs are not allowed to run free and out of control, especialy not on other peoples property. I also explained that if I EVER see his pets (pooping or anything) on my lawn again, I will not hesitate to call the pound. I did also offer to help him build a zipline for his dogs if needed.
The letter was well received, he came over with 30$ and was penitant, and even made some cookies to show he was sincere. I think that has made a big difference, and we can be civil now. Writing the letter to him really helped me express myself and get what I needed. We arent chummy or anything, but at least I can look over in their direction without feeling angry. If it helps you, a letter explaining how you feel and if needed, restitution, may help your neighbor understand how important it is to keep pets under control. Again, I'm sorry about Billy, I totally understand. Jaime