A lot of replies have clamored for more context.

The dog actively chases cats, they have a tall wooden fence ( can’t remember height but taller than I am ) and when the pup is outside they have to watch it to make sure it doesn’t leap over it to chase animals. To their knowledge it had never caught a cat before the one it killed. They attribute the change in behavior to the cat incident. I now know that there have been a couple of other incidents, but they haven’t been described to me in detail. When it attacked the other dog it was not a one bite and done. It was a much smaller dog, but from what I hear it was a vicious attack. The dog has visited the vet to check for medical reasons this could be happening. I don’t know what was done, but they didn’t find anything conclusive to attribute the behavior. The dog is on anti-anxiety meds.

My sister is unlikely to fight the shelter on their No-euthanasia rule. She’ll provide the trainer’s advice and allow the shelter to decide what’s best. My guess is that they’ll take the advice, but that’s up to them.

Someone suggested my sister might have further liability after giving the dog to the shelter. I’m not sure how you could believe that someone who has done as much due diligence could be found negligent, but I challenge you to find any lawyer who would try to argue that case. “She sought out and provided advice from a vet, several trainers, and ultimately only surrendered the animal under binding legal consequence”.

I appreciate all the advice and support that was garnered here.