I'm really sorry for our loss... The dog was just acting like a dog (no blame whatsoever), but the owner is obviously lacking intergrity and decency. All animals deserve to be taken care of in a responsible way, including this dog. I'd contact any dog rescue group in your area (just google it) and tell them about what happened. They'd be able to suggest a possible place to take the dog other than a regular shelter where he'd be in a cage, which seems miserable to me if he's a farm dog, and wait to be adopted or put down if nobody adopts him.
Humans certainly know how to be "human"... and shooting the dog doesn't strike me as that... that'd be taking "an eye for an eye". This is not the law of the jungle I think, and this situation can be resolved with good intentions and taking into account who really is the one to blame (irresponsible dog owner, of course!). Please take into account that shooting a dog is a misdemeanor in all states and a felony in many (not sure in your state).
I hope you can find a place for that dog... then you can call animal control and report this farmer. At least they'll pay him a visit and for sure he'll think twice when getting another dog and letting him roam around. I'd not call animal control before the dog is placed because they'll likely take the dog and put him down.
A neighbor's cat killed one of our pullets and although we were very sad, I didn't go after the cat, as I realized it was just his instinct.
Hope the other chickens are recovering...
Lorena