Ah... maybe frozen chicken would do the trick... ?!Sooooooooooo... don't shoot the dog. The fallout with the neighbors could literally be deadly (people are nuts).
If I were in your shoes, I would set a trap (but I'm also nuts). I'd lock up all my chickens except one brave soul who would give her life for the better of the flock- and leave a small area open in the fence faced inward- so when the dog pushed in, he'd have a serious issue getting out. Then he will be trapped in your Pen- after killing your chicken, and when you call animal control they will come take care of him.
That's obviously a bit over the top- but I've dealt with worse and come out ahead. Your neighbors laughing at their aggressive dog is enough for me to set the trap though. That dog's next victim may be the 4-year old girl down the street.