You know how to handle it. Do it. I know it sucks, BTDT, but as someone who's worked on a farm, things have to be done that aren't exactly "PC". Working farm, you're gonna have a dead one here and there and they usually just carried them out to the bone pile and left them. Coyotes took care of the rest. Just an example. Animal rights people would have had us hanged for that, sometimes I wondered if those people value the animals' lives more than ours, but whatever.
I like dogs and we value them as pets. But I'm a responsible pet owner, knowing what dogs do. They wander off and no telling what they're gonna get into. That's just their nature. We kept our on leashes and/or on a run (a long one) and/or in a pen. I don't like doing any of that but I also value others' stuff and their livestock too. We no longer have dogs now but we do have a couple barn cats and an indoor cat. Barn cats are wild, can't get within 20 feet of them but they keep the mice down and they don't wander very far off either since the food is at the shop (I feed em about twice a week). The indoor cat is just a pet obviously, laying on my chest as we speak.
We have a new neighbor who's got a pit, and it's not all that friendly to us. Wife won't go outside, darn dog roams around and when she goes to get the mail and the dog is close, it growls and barks at her. I told her to keep an eye on it and take some video of it. I hope she doesn't have to put it down but is so, so be it. She's smart enough to know when it's time. Have no use for those dogs, have had them and they are worthless for anything but hog hunting. One of them tore a lot of the scalp off of my brother's head when he was young, no reason that I know of just random. Then he had a shephard attack him too a number of years later. So it can be any breed; but some are a LOT stronger than others which is why they are good at what they're bred to do. Mom's got a ACD (blue heeler) which is a great dog but hyper and can bring the blood when she's nipping at your heels, but that's what those dogs are bred to do-to nip at the heels of livestock to herd them.
S.S.S if you have to.