Some cats DO like to be petted, but after a short time it is too much "excitement" for them, and they will bite to tell you to stop doing it....some cats are VERY sensitive about being petted, especially on the stomach, and won't stand for much of that at all...in that case it's best to just pet them on the head, face, under the chin, around the ears, all those places on the head that you KNOW they like.
He shouldn't be biting you or your family enough to draw blood. Have you talked to your vet about this behavior?
Cats are good at curling up very tightly to keep warm; or getting under blankets or towels, or whatever they need to do to keep warm.....
I don't think a cat muzzle (which to my knowledge doesn't even exist) is a good idea at all....try to be more in tuned with when he's telling you to STOP petting him!!! Just because he "seems" to want to be petted doesn't mean you HAVE to do it! Try some games, like "Da Bird" where you play with him with a thing.....looks like a light fishing pole, with a string and feathers attached....you throw it, holding on to the base of course, and he chases it; you can make it move in the air, or "dance" or whatever to entertain him......do 15 minutes of that a couple times a day, which gives him exercise and bonding with you, and is very entertaining for the cat.
Can you brush or comb him BRIEFLY every day for under 5 minutes, no more, because too much is too exciting for him, and will lead to biting in his case, as that is what he has found "works" for him to make you stop doing things he doesn't like....you need to CHANGE THAT BEHAVIOR!!! If you can comb or brush him, briefly, every day, the mats shouldn't build up like that have, requiring him to be overly sedated for the vet to shave him-----THATS not good for him at all. Maybe you need to look for another vet? I'm surprised your current vet hasn't had any better suggestions for you about all this, and that he's routinely "putting him to sleep" to shave him.....thats just NOT the best management for the cat.
Good luck with him, he CAN learn to change if you work at it.