Our DD has been a tummy sleeper since she was 5 weeks old. She would not sleep on her back. A week after I flipped her to her stomach, she started sleeping through the night. Like from 9:30am to 6:00am straight. We do not use the Angel Care Monitor, but do have a video monitor and she sleeps in our room. I would talk to your doctor about it, if you don't feel entirely comfortable putting him on his stomach.
My opinion is, yes sleeping on the back is better and safer for the baby...but sleep is important for the baby. So if sleep isn't happening because he or she is on their back, it is your decision to make as you are the parent.