I am so sorry for the trauma you and your family has had to endure.
I want to put in my perspective here. I live in the country on 80 acres. My neighbors all have horses and cows (including a dairy farm about 2 miles down the road). I have 4 horses, 1 mule, 6 DOGS (got to love them or I wouldn't have that many) that are mine plus one stray that showed up that will be going to a new home next week, and now the chickens. I also have a cockatoo, a cockatiel and 30 sugar gliders. Each one of these animals have a special place in my heart and home.
Where I live, if there is a stray dog and I call the SO (there is no animal control office here) and tell them about the stray, their response is to just shoot it. They can't/won't be bothered. I have a soft spot for strays, always have so I will do what I can to find them a home. HOWEVER....let that stray go after ANY of my other animals and it's a dead dog.
We also have coons, opposums, bobcats, and coyotes (as well as deer, turkeys and a huge assortement of other animals) that are part of life here. We accept the risk of some of these being preditors to our "captive" animals. BUT, both me and my husband are fully prepared and willing to stop the threat to our animals at all costs.
That being said, this dog was NOT a wild animal but a pet of someone. That pet probably meant a great deal to that family and I'm sure they are hurting from their loss. HOWEVER! Your daughter's safety was at stake. Any type of dog in a blood lust is a dangerous animal. And it doesn't matter if it is a 200+ pound Bull Mastiff or a chihuahua.
My neighbors all know if MY dogs come on their propery and are even just harrassing their livestock, they have every right to destroy my dog. Just as I would do to theirs. It is your responsibility to protect the animals you bring into your life. And more so to protect your daughter's safety.
There shouldn't be any doubts about your actions. What was done was completely necessary. Who's to say their dog wasn't rabid? Was it sent in for testing? What if it was rabid and once done with the chickens did turn on your daughter, you or your husband. And for them to now be harrassing your family? NOT ACCEPTABLE at all. Time for law involvement only because "country justice" can get really ugly.