Does anyone co-sleep with a tummy sleeper? Is it safe?

LO has cut out his first night waking but has started waking to eat at 5:30. He clearly isn't ready to be up for the day at that point but if I put him back in his crib afterwards he cries and wakes himself. Today, I brought him into our bed and side nursed and he fell asleep for another hour. The problem is, he kept trying to roll on his belly. I couldn't sleep because I was so paranoid he'd suffocate in our soft, tempurpedic mattress.

Is he ok on his tummy in my bed or should I keep trying to roll him back on his side/back?

I'd prefer he go back in his crib, but I'm not sure how to make that happen!