Hi Molly,
How big is his cage? Many commercial rabbit cages are simply not big enough, so he probably just doesn’t like being confined to a small space after having free run of a room. It should be big enough for him to do 4 hops across, stretch out in width wise, and stand up to full height in. He should also have at least 5 hours out of it a day.
Is there a way you can block your bed off so he can’t get to it if he was free range at night?
Have you tried giving him a stuffed toy to cuddle with?
The other problem is that you’re reinforcing his negative behaviour, so now he’s learned that if he nips and burrows at you he gets what he wants, so he’ll keep doing it.
He does sound like a bunny that might benefit from a bonded partner, since he sounds quite attention seeking. Of course, this is a big commitment and its certainly not the only solution, but it could be something to consider. There’s lots of factors to think about if you did go down that route, and people here could advise.