The Neighbor / dog thing is universal and irritating, to say the least.
Making a nuisance of yourself with local law enforcement, is not an easy way to go. While pledged to uphold the Law, you will quickly find that the recognition of animal ownership is all in the dog owners favor, Puppies are cute and who cares about some dumb chicken. You will find ample resistance to making a dumb neighbor, for ego or whatever, to toe the line.
A pellet gun, at range will not break the skin, but the sound and sting will be a good message. Unfortunately, dogs are smart too, and will only avoid YOU, and the uncomfortable threat. A paint Ball marker is what I used once. The animal returned home with spots all over. The balls leave bruises on humans, so I assume the equivalent pain for a large dog. Again, range is short, and the dog will avoid YOU.
I believe a letter and a copy in file is the proper approach, farm dogs are normally given free run, and some get an overbearing taste for chicken. I have two dogs, a Great Dane, and his little Buddie, a Sheltie, neither of which range far or hunt for themselves. They are trained to not leave the unfenced property, of about 5 acres. Never the less, if the final option, of .22 pellets for the interlopers occur, you open yourself to retaliation, on many levels. Think carefully, about your plan, unless you like private and personal warfare, that is largely outside the given law. I have to say a few chickens lost through time is inevitable, and may not be worth the fight. However, being an Alpha dog personality, my showing up on a doorstep with a problem, usually goes a long way to a solution. Practical law, as used in country and urban settings, is simple. Don't molest your neighbor, and the law is administered to keep hostile factions apart. That is why a reasonable approach first is the best bet, until all alternatives are exhausted, before the final solution.
This subject is a recurring and universal complaint on this board, and there are many opinions, both rude, and thoughtful. so I will comment no more here.
This thread was very informative for me. I use bag feeds of several types and 'bird food', from the Home Store for variety of treats and natural Milo, Millet, and Sunflower. They run sales a coupla times a year, so it is less than bag feed in the larger lots. I have a bread store outlet for 'Animal bread' that is only $2.35 per bag that has an average over #20lbs each. They are not for human consumpion, and bagging is peirced, you must sign off on a list. But we consume about a bag a week as treats for the flocks, throwing about 2-3 bags of buns, breads, per round. It is well liked and causes quite a stir in the pens, and has the fortified vitamins for humans in the mix. The only thing that must be guaded against is mold in higher summer temps, as the breads dont keep well in heat that makes it sweat in the bags.So we dont 'stock up', just grab a bag catch as can, and feed it off.
My bag feeds cost me about $13-14, with discounts for over 5 bags. So that comes close to $700 per ton, on store bought. Corn here is $9 a bag, $9.39 cracked. I had been thinking of a garden type shreader for running grains and material through, ( hammermill type), but it seems too small a gain, and intnese labor and transport to achieve a minor goal. So that idea is shelved. I am not into buying more work. Hoisting a bag or two is bad enough at my age.