M.P.
I suggest that there is nothing wrong with your son. It sounds like he has a realistic view of the world of animals and science. Your husband has to have a removed attitude toward an animal to dissect it. He can't think of the rabbit as a pet. I suggest that your son has picked up that attitude. You probably talked about eating rabbits.
I suggest that being honest with him as in saying that you're skinning the rabbit would make it less confusing. I don't see you son saying you're skinning him as being cold. He's being factual.
Is your son not warm and affectionate with you? If not then I'd be concerned. But being realistic about a rabbit sounds reasonable to me.
When I was growing up my family ate rabbit. It's still one of my cousin's favorite meats. We raised our own beef. My parents talked about the difference between a meat animal and a pet. We were not allowed to get attached to the steer. I suggest your son saw his father skinning the rabbit and he didn't get attached to the pet rabbit. That seems like a healthy response to me.