He’s marking his territory when he pees and poops on your bed, some bunnies also just really loving to pee on soft things. Can you limit him to floor time without letting him on the bed? Or does he just jump up anyway?
Neutering would certainly help with his litter habits as it cuts down on territorial marking, however no one can say for sure that it will 100% fix it. It may take a bit of work because he’s probably quite in the habit of going to the toilet there now. Neutering, while scary, is very routine surgery and not invasive. If you are particularly worried about it, you can request to get a blood culture done before hand (extra $$) to check his organ function etc to make sure he’s healthy for surgery. Bunnies are dropped off and picked up the same day and don’t need to be fasted. I was really worried when Bandit went in for his snip, but nothing went wrong
Does he use his litter tray at all? Perhaps he doesn’t like the position that it’s in?