OP has been speaking with their law enforcement. They have already shot a dog but since they and their young children live there and are planning to continue living there for some time, they don't want to cause unnecessary issues with the neighbprs. Hence why he hasn't wiped the pack out already.
Everyone chooses their own actions, based on what is most important to them.
They can also swear out a legal complaint or warrant. There is probably no fee for doing that, and law enforcement might be sympathetic. MO laws might hold that anyone can destroy another's property without consequences, or that the owner of a dog might not be responsible if the dog attacks livestock or people. I seriously doubt that, but it's possible. Sometimes the law really is an ass.
OTOH, the dog that killed my chickens looks like a cross between a lab and a Great Dane--well upwards of 100 lbs. He also jumped over our fence earlier in the year and was standing a few feet from my back door when he saw me out by the chicken coop and started barking at me, then started running at me in attack mode. I picked up a sizeable rock and started running at him. I let fly when he was about fifty feet away. I missed him, but it made him stop and change his mind about whose yard it was. I
should have gone to the owner then.
My kids have flown the coop, so I'm not as worried, and I do know something about dogs. But if I had small children at home and a predator who might want to consider the place of his food source (my chickens) a place from which he would want to drive off competitors (small children,) I wouldn't be concerned, I'd be terrified for my children.
Dogs are very territorial, and have to be trained not to be dangerous to children. As a boy, I delivered papers when there was no collar or leash law, and got bitten at least a dozen times. Most dogs are fine. I like dogs. But they are pack-based territorial predators, and we should never assume that Fido is safe towards a strange child because he is friendly with his family.