I own 2 dogs, a lab and a Chessapeak Bay retriever. The chessie has a very strong mind of her own and likes to run. Occassionally on the neighbors property. She has run up and smelled the horses over there but never bit anything. However, the neighbor did tell me they will shoot her if she's over there again. She hasn't been (thank god puppyhood is over) but if they did shoot her I would completely understand! It is MY JOB as a pet owner to confine/restrain/train this dog.
having said that, I would not hesitate to shoot a dog in my yard. And I'm not shy about letting the neighbors know that. I have 3 goats, they are not fancy prize winners, they are my weed-eaters. They are kept fenced to protect them not contain them. One was attacked by a dog when I wasn't home, It got through the fencing that I put up to keep her safe, she was bitten but only once and in the front which means she didn't run away but stood and fought. A goat head hurts when they butt you like they mean it. I was very lucky, I have many firends who have come home to their expensive show or dairy stock ripped apart all over the yard. I don't know whose dog it was, so I can't go do anything about it but pay the vet.
My suggestion would be to tell the neighbor A) You know her dog did the damage. B) Here's the bill for the fence and the replacement of your bird. C) We will shoot to protect our livestock and D) File the complaint and be sure you have animal control inform her you have the right to defend your property.
I wouldn't worry about disrupting the peace in the neighborhood because it has been my experience that there really isn't any peace when a dog or dogs are running free. Wether they are eating animals or just pooping on lawns, only the owner of an unruly dog has any real love for it.