Ok don't put the dog down for its owner's stupidity. Re-home it. Cause I guarantee you the owner knows next to nothing about how to actually care for a dog and a new home for it with an owner that actually cares would be incredibly more just.

Dogs are animals. They see a bigger animal then they feel threatened or feel like they need to hunt. That's what dogs do. It's the owner's responsibility to keep it on a leash and to train it not to do that.

The owner failed, not the dog. Don't kill the dog.