Waking up "with the birds" very early morning is normal for cats, and they of course want their people up with them. Cats are also creatures of habit so you sure do want to not let biting get to be a habit.
Getting a "stuffie" (I like that term) would be good, but your kitten might still prefer you. You could try putting the kitten under the covers next to you, snuggling it closely to your body and petting it some as you drift back to sleep. That worked for me with one of my kittens who wanted early morning attention. My hand was then in a comfortable position to keep petting while I was able to fall back to sleep. He didn't bite though. He did come to love being all warm and cozily snuggled up against me, cradled with my arm. He would start purring and kneading. (He still meows for me to lift the covers to let him under, now at three years old.) Maybe your kitten will be satisfied with the warm snuggling against you, even if you fall asleep again and quit petting. Maybe the kitten will fall asleep too! You can hope. Just be sure the blankets are loose at the top so the kitten can crawl out if he gets too hot or it gets too stuffy under the blankets.
If the kitten is less then 12 weeks old, I would *not* let it under the covers as I would be concerned about suffocating it, it being too small, fragile and weak to fight off covers or move out from under a person about to roll over. My kitten was about 6 months old when he came under the blankets.
If the kitten starts to nip your hand while you have him under the blankets, (or in any location) you can just put your hand on his back gently, to keep him still and give a "I'm dominant, so stop it" message. Mother cats will sit on top of kittens who get too rambunctious or bite, to control them and make them be still. Your hand will remind the kitten of that.