Keep putting baby on her tummy - if you're worried about SIDS, please rest assured the back-to-sleep campaign hasn't reduced SIDS deaths one iota - sleeping position has NOTHING to do with mysterious crib death.
All three of mine were tummy sleepers from one hour from birth on, and slept extremely well. Back-sleeping is simply uncomfortable, aggravates gas/reflux pains, and creates a startle reflex with loose limbs. Please don't ever use sleep positioners - those are suffocation hazards and need to be removed from the market.
EDITED TO ADD: If a monitor will help YOU sleep better, spend the money on it, however, please be advised a SIDS baby cannot be revived... these monitors do NOT detect SIDS. Please also keep in mind, SIDS is SOOOOOOOOOO rare (you have a higher risk of injury or death to your child in an auto accident), that frankly, it's not worth thinking about AT ALL.
Tummy sleeping will solve your problem. Good luck!