First of all, this is just a season of your life and it will pass. Keep in mind that this will not go on the rest of his life! Sleeping on his tummy is dangerous whether you check on him frequently or not, so I'd suggest you stop that immediately. It's possible that he's gassy or uncomfortable sleeping directly on his back, and it's possible that he just needs to change positions often. Have you tried his car seat? Then after he wakes and nurses, maybe in the swing, then the next time, in the co-sleeper. Maybe he's to hot or too cold. Sleeping only 2-3 hours at any given time (day and night) really is par for the course at 7 weeks. You can't expect him to sleep all night, or even longer stretches until abuot 3 months in my experience, and when nursing, most of mine didn't sleep though the night until closer to a year.
I just read through the rest of the responses. I just had to say the SIDS risk is real. I just read an article on it at the Dr.'s office. I know for me, that was not a chance I was willing to take. Just like I wouldn't put my child in the car without a car seat. If, on the off chance, something happened, I would NEVER forgive myself for putting my child to sleep on his stomach before he was ready. I'm not trying to be an alamist, but I think so many people are like "that is so rare and my kid is fine" but if it did happen to you, you would be devistated!