I have a four-year old and a four month old. It is not a risk I would take. Clearly, you aren't even trusting your own instincts on this, or else you wouldn't be so anxious and still letting her do it.
I am sure you aren't the first mom to feel this way or who gives into temptation in the wee hours after trying to get a screaming newborn to sleep. But the sleeping on the back thing, is the one line of research which everyone agrees on when it comes to SIDS. Keep the baby on her back for one year and nothing in the crib (toys, pilow, etc.).
When they are able to roll over on their own, there is not much you can do, but until then it is up to you to give her the safest possible conditions. If she really is intrinsicly a tummy sleeper, then she will probably roll herself over in a few months, so you don't have long to go. But in the meantime, you have many nights of worry ahead of you, which I have to say would be justified worry.
I sympathize with how you want to make her comfortable and get her to sleep. But, as an experienced mom, you know this is the first in a long line of things that you need to do for her safety, whether she likes it or not, and whether it is easier for you. Also, you know how quickly they change their sleep habits. I think a few nights of back sleeping and she will probably be used to it. Babies this age often have difficult sleep behaviors, that is just how it is.
Lots of moms make compromises for naps - like putting the baby in a swing or bouncer - I did this too. It's fine as long as you can supervise them. Maybe, when you are awaks for her naps, one of these or the tummy even might be okay. However, everytime you put her on her stomach, you are reinforcing the habit.
Congratualations on your new addition - remeber it gets easier soon! I am just two months ahead of you and my back-sleeper sleeps 10-12 hours at night.