Well, like normal people I don't carry a shotgun around with me while doing my weekend projects
Well, true, I too find a shotgun a bit noisy and indiscriminate
Welcome. Sorry to hear about your girls.
I'd just like to expand a bit on what has already been suggested:
1. There is no substitue for knowing your state/county/local laws (check the statutes/ordinances/etc. and then call those agencies responsible for discharging those laws). Make sure you get the name of the person you speak with - if they'll let you record the conversation do so, otherwise memorialize the conversation in writing immediately (the person with the `best' story, i.e., most accurate, usually takes the day). Find out
precisely what actions you are allowed to take. If the situation goes south and the cops are called, I can almost guarantee they will not be able to quote the law (you will), and you'll have documentation.
2. Maintain a complete record (with receipts) of everything purchased for your flock. Depending on the laws in your state, it might be possible to intiate a civil action to seek damages for the neighbor's
depriving you of the right to use and enjoy your property.
Whereas the most you might receive in small claims court will be the replacement cost for the birds
3. Video surveillance is getting cheap, even an old CCTV camera (doesn't have to work) placed under the eaves and aimed at the backyard (in a position the neighbor's can see) is intimidating.
4. Do not discuss anything with the dog's owner, in person, without a witness and your pad of paper and pen. I can say, from personal experience, this makes the other person very nervous (be it County offical/cop/dog owner).
I wish you the best of luck and, again, sorry about your girls. Stick to your guns and, eventually, everyone in the area will realize that your place is not a KFC where the meals are served up alive and screaming to sate the appetite of, and provide amusement for, their vermin.