I'm very sorry for your loss, that sucks.
This is just my opinion, so take it whichever way you want. And you're pretty much in the right no matter what, this is just what I would do:
Get to know the story first. Shooting someone's dog is big business, it can lead to a lot of heartache for the owner and a LOT of future trouble for you. There seems to be an idea going around that if you see a dog roaming freely that automatically makes the owner irresponsible and it means they don't care about their animal. I don't think that's the case, I know dogs that are just straight up escape artists, no matter what the owner does to stop them they find a way. What if the dog escapes because their clueless friend left the gate open? What if the dog gets out because they were planning to build a better fence 3 days later? What if this was the first time the dog got out and the owner's haven't even realized it's gone yet? Maybe these are unlikely, but it's still
possible.
I wouldn't immediately shoot the first time they attack (depends on what it's doing though, like is it chasing the birds while they free range or is it in the run wiping out everything? If the latter, I would shoot immediately.) unless your birds are priceless to you, like if they're extremely important breeders or if they are your primary source of income or something. But I would try everything to scare it away/apprehend it. Then I would try to see if I could to contact the owners, and tell them what happened and see if we can work it out. If they apologize, offer to compensate, and promise to make changes I wouldn't shoot the dog even if I saw it again once or twice, because
they're working on it. But if it keeps happening, and it's clear they're not doing anything, after lots of warnings, I'd shoot.
If you couldn't contact the owners whatsoever, or if it really did seem like the owners didn't care. I would also shoot.
But that's just me, and I can be a bit too forgiving sometimes. I'm sorry people are so upset at you, we put dogs on waaay to high a pedestal these days. Best of luck.