I had the same experience you've had at the beginning of my pregnancy; I was taking lovenox and all the midwives said I needed a high risk doctor. It took me a while to accept, but I finally ended up going to Dr. Mussalli and Dr. Worth at St. Vincent's. They don't have a birthing center, but I've been told they have some nice birthing 'rooms' (haven't toured yet). St. Vincent's has a reputation for being very midwife and doula friendly, and Dr. Mussalli and Dr. Worth's website is also doula friendly and open to more natural childbirth options. Dr. Mussalli is the high risk doctor, and he is also the head of ob-gyn at St. Vincent's. In the end, he looked at my bloodwork and decided I didn't need the Lovenox and was only high risk because of my age (37). In the end, I decided to switch back to a midwife at St. Vincent's, but it was a hard decision, because I really did like Dr. Mussalli and Dr. Worth. I also spoke to a doula who had worked with Mussalli and said it was the first time a doctor had ever asked her advice about things! Their website is
Village Obstetrics & Gynecology | New York City.