Doc,
I doubt I would have the resolve to do this, unless it really went after one of my kids, especially having two dogs myself, but around here they call it the 3 S's.
Shoot, Shovel, and Shutup. And if your slick you take the colar and throw in the road on the other side of town.
I have had dogs kill chickens and even the occasional lamb. Sometimes I even know who's dog it is. This fall I returned a dog with blood all over it after I watched it take out one of our freerangers and got only a "sorry". In the interest of neighborly relations and the fact the hen was past her prime and the rest of the family wasn't around to witness the carnage, I didn't follow up with anything but I know it is only a mater of time before it comes back.
Unfortunatly this just sets up bad blood. I think the owners should be more responsible, but they just think their family pet is having fun. Boys will be boys kind of thing.
But make sure if you hit the dog, you kill the dog. One guy I heard of got sued and lost when the wounded dog returned home. But the only reason the people with the dog knew who shot it was because their dog had recently mauled some of the farmers sheep and he had done the "right thing" and returned the dog only to see it come back and try to do it again. Interestlingly the judge determined that the farmer would have been in the right IF he had actually killed it as he was protecting his property but because he had only wounded it he was liable for damages.
On the bright side it would be a good reason to buy that backhoe you have been wanting, a cheap alternative to a lawsuit, but on that note I say spring for a used mini excavator instead. Not much more money than an installed new hoe, but many times the digger... a hydrolic pump on wheels