Sorry for your loss. Is the dog's family taking steps to prevent it's future escape? I wouldn't kill the kid's dog at this point either, but I would make sure they understand that their dog killing your kids' pets isn't exceptable. Because you say that when he goes out to feed it, the dog bolts out the door, I assume that the dog lives in some sort of kennel or out building. Perhaps this family needs to get out and pay more attention to the dog. Most dogs that sit alone all day get bored, and nobody can blame them for wanting/needing to escape at every opportunity. The child and the dog are innocents, but the parents, regardless of their remorse, may need to take a proactive measures. Dogs need a lot of attention. I hope they work with their pet. It will go along way in keeping yours safe.
There is a couple that live near us that are the sweetest acting people in the world. Years ago, their Husky started getting loose. First it went for my daughter's rabbit. They were apolgetic and swore they would fix the fence so it would never happen again. They said that the dog meant so much to their daughter they couldn't bear to not let her keep it. They were so sincere and sweet we forgave them. Then the dog got loose again and got in our tack room and chewed up a $100 bridal. Again they were so apologetic and swore that this time they would fix the fence even better. I was mad, but for the sake of neighbordood peace I forgave. It got loose again. This time it had their new dog with it, a Rhodesian Ridgeback. They were chasing the horses. OK, now I'm really pi$$ed. They came and get the dogs, and I read them the riot act. The apolgetic parents and the crying daughter were begging that the sherriff not be called. No animals were injured so there were no damages. Guess what, it happened again. This time a horse was bitten. The dogs attacked it and were trying to bring it down. I didn't own a gun yet (I have many now), but I chased the dogs off and called the sherriff. The daughter took the dog and hid with it. The parents weren't home. The deputy couldn't find any of them. Then I got a phone call from the kid, and she was swearing like a drunken sailor and calling me every name in the book. The dogs have not been on my property since, however, they, and the 6 Rhodesian Ridgebacks that they have since acquired, have made trouble for other neighbors. They killed a neighbor's Pomeranian, beat up a mixed breed ($1500 in vet bills to save it), and attacked a Lhasa Apso. Each dog was in it's own yard when attacked. Each time the sherriff was called. Each time they got a ticket. Each time they paid damages to the owners of the injured/dead animals. Each time the city/county attorney let them off the hook. Each time they were pitifully remorseful and swore it would never happen again.
The moral of the story is "Apologies are only worth anything if the offending party actually takes steps to correct the situation." I hope your neighbors are more than just nice. I hope they are proactive. As for me, the gun is ready. It will be SSS (Shoot, Shovel, Scream as loud as I can - I never was good at shutting up!)
Enough of the rant. I hope your neighbors ar truly good and the trouble is behind you. Bless you for being patient and kind, and I hope your birds are doing well.
JMHO