I am a home day care provider, and I therefore cannot have a baby sleeping on their tummy in my care because of the SIDS risk. However, my own kids have slept on their tummies ever since about 3 months old when that was the only way they would sleep. I could go on about why I chose to go with it for my kids, but I won't right now. :) That was also around the time they didn't stay swaddled anymore and their startle reflexes would wake them if they did fall asleep on their back unswaddled. I think a lot of the comfort babies find in sleeping on their tummies is that comforting pressure on their bellies and arms, kind of like they are hugging their bed, after being cuddled so tight in the womb, it makes sense that cuddling and that pressure that goes with it would comfort a baby. Swaddling helps that feel for them I think, like the blanket is "holding" them. Lying on their backs free gives kind of that free fall feel instead. Also, sleep begets sleep, so if she is getting overtired, it's going to be harder for her to fall asleep. Sleep is so important for health and immunity, growth, brain development, etc. I would try swaddling for naps too, so she doesn't have to sleep on her tummy and still feels comfortable enough to sleep. Also, the Angelcare monitor that somebody mentioned. Sounds good in theory, but as they said at the Sids seminar I went to, "it only tells you when". Even if you are right there with them, it's not something you can just wake them up from.
Just some thoughts. Hope it helps.