I'm in exactly the same position, with an 8 day old ds. I suffer from reflux myself and wouldn't leave him to cry because I know just how painful and uncomfortable it is to lie flat.
Lots of good suggestions above, and here's what is working for me (although I'm still sleep deprived!):
Raise the moses basket to a 45 degree angle. During the day when you are able to keep a very close eye it is ok to put her down to sleep on her side or her front. This was a tip from the midwife. My ds is able to sleep much more comfortably that way.
Don't be a victim during the day. Put her in a sling and go out and about, meet your friends, get your jobs round the house done. A tiny baby in a sling isn't much of an obstacle. You can even sit in a cafe wearing the sling, provided you check she isn't over heating.
Co-sleep as suggested above with her on her side, head on your arm. You'll be uncomfortable and probably won't sleep yourself, but you will be able to close your eyes and rest, and possibly transfer her to her moses basket once she is asleep.
Smaller feeds more often. The midwife watched ds feed and told me he was a guzzler, and would sleep easier if his stomach wasn't completely full. She was right. I liked this midwife, she was old school, not particularly 'correct' in her advice but with years of experience. Her advice was do whatever you need to but be vigilant.
It's tough, I had to hold ds upright for 5 hours straight the other night, so you have my sympathy.