Before you do that......

(Ugh, I can SO relate to young Hardy....I am an animal person)

It could very well be that the dog was going by instinct to chase (and eat) small game....what kind of dog is he? How old?

If he is a ratter, and has never been around a kitten, it may have just been a case of mistaken identity. How old / mobile was the kitten?

Snapping at BH is another thing....does he have a history of this? Is he protective of his food or toys? Sounds like he is showing fear-aggression with your man (with good reason, if he got smacked). If he has never done this before, and you want to keep him, BH may have to take over being the one to feed him and give him positive rewards for a while, to gain the dog's trust back.

Please don't throw the dog out for that, if you can. If he really has become a danger, you would just be passing the problem along. Might be better to put him down, if that is the case.