I always thought I could kill a dog running deer in my zone but can't help but think someone cares for that dog as much as I care for mine. Heck I would give up the wife and kids first b4 I gave up my brittany.
Granted, I don't live in an area where feral dogs are a problem but would sure have trouble shooting a collared pet. I do keep a rubber blunt in my quiver and think that might get their attention with a shot to the rump but I doubt they would associate that with running deer and would probably be doing it again real soon anyway.
If you do lace a dog just don't let the word get out. Could be a real black eye for hunters.