I think we may have lost the OP, but I'll add my 2 cents' worth, anyway. Electric fence is not terribly expensive. A roll of wire, some wooden corner posts and step in or T-posts. The thing to remember is, if the dog can get to your chickens, so can any other ground predator. (It's the price we pay for free ranging.) If you don't want your chickens to get eaten, put up a fence.
That being said, the neighbors don't get a pass for being irresponsible. I also live in an area where it's understood that if my dog kills or harasses your livestock, it will likely get shot. (Often, the dog's owner will take care of the problem themselves in order to be a good neighbor.) If you shoot my dog because it's been killing your animals, you have every right to do so and it's entirely my fault it had to be done.
If you know who the dog belongs to and feel it would be effective, talk to them. If you don't feel that would be effective, catch the dog if it's friendly and paint a bullseye on it before sending it home. Or, as stated previously, make it disappear however you feel you can. If you need to kill it, do it in a way that it doesn't suffer.