I just finished reading the thread. When I first started reading, I will admit that I wasn’t that concerned about the dog killing other animals on HIS turf. Those other dogs were in his territory. Honestly, I still feel that way. If he were going after other dogs while on leash or getting out and hunting them down, I would have a bigger problem with it. My dog kills rabbits, squirrels, raccoons, opossums and stray cats. Anything that comes onto our property. He is part coonhound and from what I have read, that is a territorial, protective breed. I don’t know what he’d with a small stray dog.
That being said, I do understand OP’s concerns. The kids come first. Always. She did reach out to a rescue and even they are unwilling to take this dog. I agree that putting it down is, in this case, the right thing to do. If it’s done humanely (and I have no doubt the OP will see to that), the dog won’t suffer and will have no idea what’s going on.
OP, it’s a tough decision, but you have to do what’s best for you, your family and your situation.