You should be cautioned about trying to solve a dog problem with BB's, pellets, rock salt, paintballs or any other purposeful, nonlethal means meant to mark or cause pain to an animal. The laws in most jurisdictions allow you to humanely kill a predator dog, but they do not have allowances for marking or causing pain or injury. You can be charged with animal cruelty for using such non-lethal methods. It may seem odd, but it is often legal to kill the dog, but illegal to mark or injure it.
Personally, I would shoot the dog and demand payment for damages. There was a time when I would have not been able to shoot a dog. Then my neighbors with the rhodesian ridgebacks came along. Those idiot owners and their idiot dogs changed my perspective. It's not hard to convince a judge about the cost of raising healthy birds. A judge awarded me over $1000 for 13 chickens killed by my neighbor's dogs. If animal control is useless, call the sheriff or the police department the next time the dog is loose. Keep calling until you get resolution. Don't let up on AC either. Be the squeaky wheel. Demand action. Be a pain in their butts. Demand that the people be ticketed. Check into the ordinances and see how many strikes they can have before the dog is removed or the fine increases. Make sure that the authorities are holding them accountable to the maximum punishment.