Ok, I am on both sides of the fence here...and maybe that will allow me to be unbiased, I hope.
First, I think it is a great idea to try to keep the hostility factor out of this. You have to LIVE by this neighbor, and a bad neighbor can make your life miserable. Kudos to you on keeping it calm.
NOW, I love my chickens, and I love my dogs. I made the error of assuming my dogs loved the chickens too, since they have spent 9 weeks cohabitating in peace. We live on a farm, and I am one of those "irresponsible pet owners" who lets their dog loose, all day. prior to our owning Tim, he ran off daily, for 3 years. We got him and brought him into our home as family, as opposed to being in the kennel like he had lived, and he never left again. In the nearly 8 years we have had him, Tim has left the property once, during a thunderstorm that freaked him out. The cops returned our 120 pound wet, stinky Malamute in the front seat of the patrol car. We have since been careful to make sure if there is a chance of rain to bring him inside. He is inside the house 75% of the time as it is, but he enters and leaves as he pleases. I have considered him a plus to have with the birds as he slaughters any furry predator on sight. Seven years ago, he was a notorious cat killer. We tried everything, and at the end of our rope and having witnessed the murder of four baby kittens that we THOUGHT were securely locked up...we did the unthinkable...well for us anyway. I hesitate to mention this lest I inflame a serious animal lover...but seeing as how everyone else is talking about shooting the dogs, I think I am ok. My husband recalled the advice of an old farmer who had chickens that were killed by a dog-and he decided to use it. Hubby grabbed the dead cat and beat the living tar out of the dog with it. This was serious wooping...if you can consider a dead cat a weapon. There was yelping and growling involved...not just on the part of the dog. He also grabbed him by the scruff of the neck and gave him a good shaking while yelling, "NO! BAD DOG!!" Now, the farmer also said to tie the chicken (or int his case, cat) to his neck for several days, but that was seriosuly unappealing, and ultimately, proved unnecessary. Tim never so much as blinked at a cat again. They now live in harmony, eat from his food dish and curl up beside him to sleep. He does not love them, but he accepts them as part of the family that he is not allowed to eat. Now, yesterday, Tim was once again overcome with blood lust. We spotted him tearing into a Speckled Sussex. Stupid us-we let them free range and since we have gotten into the habit of trusting Tim, we never thought not to. For several weeks he has ignored them-now this. So, my husband recalled the farmers advice that worked so well in the past. He pounced on Tim while he was eating the chicken and re-enacted the cat scene. It was pretty hard to watch. Later, he took Tim on a leash out to the chickens to see his reaction-Tim refused to let them come close to him and shied away. He was then brought into the hosue and showered with love and kisses.
Normally, I am 100% opposed to hitting a dog, under any circumstance. BUT...you have to give dogs credit for intelligence. Just like an 8 year old kid that plays with matches despite all the times you warned her and despite the fact that the last time she did it you made her look at pictures of burn victims.... ahem.. might NEED a serious spanking to ensure it NEVER happens again...sometimes, a dog might warrant physical discipline. Now, I am not saying every dog would respond to it the same as Tim, he is quite exceptional in many ways...but...a dog that has respect for his master very well might. Remember, in the wild, wolves are not gentle with each other. The Alpha will very severely discipline an errant wolf. They DO understand authority, and will respond to it if it is used properly. I do think it is important that it be done only at the time of the incident-beating a dog after the fact will only confuse it and make it scared of you. SO....my suggestion, as I hate to see a dog get shot....and I would hate to have someone threaten me with shooting my dog...is to ask the neighbor to bring the dogs over, one at a time, on a leash. Using caution, let them approach the chickens. Perhaps keeping them behind a fence would be your safest bet. At the first sign of aggression, have the neighbor discipline the dog. I would have him repeat this daily, until the dog ceases his behavior. Make sure that whenever the dog backs off, the owner praises him profusely. Now MAYBE it won't work at all, maybe what people say about a dog killing once will make it always return to the source...but maybe not. Obviously, Tim is on probation, and it will be some time before we trust him loose with the birds again. But, knowing Tim, and the type of dog he is...I dont doubt that he will learn they are off limits. Most likely, the neighbor loves his dogs the way you loved your chickens, and he will want to do what he can to keep them from getting shot. he will probably tie them up in the yard alld ay, thinking that it is in their best interest. They will, inevitably, at some point in the future, get loose, and come back to your house. Or he will let down his guard, thinking they wont leave again. Besides, I know I would not be able to shoot a dog...unless maybe it was attacking my kids. I have owned alot of problem dogs over the years...and with one exception, we were always able to get them to break their bad habits...our Pug Sophie came from a puppy mill and had serious issues...but the change in her has been marvelous. Dogs respond to love, praise, and authority. My mom always said, "You get more flies with honey than with vinegar." This could hold true with both the neighbor and the dogs. Sorry for the windy post, but I hope it might help. I wish you all the best-please update how things work out.