Point I made earlier needs to be addressed before too much effort invested in shooting dog. Where the chickens on the chicken owners property? Best I can infer from OP's post is that multiple neighbors are present and that chickens are free-range kept. I am betting OP's yard is only a couple...
This is the better long-term approach. Law enforcement and neighbor sentiments will be reactionary only in most instances. You will also have other dogs and wild predators eventually as well. Make certain those chickens are staying on your property, if not this discussion is a waste of time...