Who has used a birthing center in Florida?

I was planning on using Heart 2 Heart in Sanford. It is close to home, hospital, and my insurance told me they would cover 100%. It was a no-brainer. I went for a tour and consult, and everyone seemed lovely and I liked the facility. My initial visit is next week, and I called my insurance today to verify what I had been told previously. I was told that they didn't have any in-network approved birth centers in my area, and that if I wanted to use a birth center, that the center would have to submit a request for authorization and maybe they would approve.

I called Heart 2 Heart and the billing person looked into it for me. She called my insurance company and got them to verify that they DID cover me and that all I would owe is a $40 copay on my first visit - beyond that they covered 100%. Then the billing girl says "Now, you were told at your consult about the $200 registration fee right?". Ummm, No?! She then explained they charge a $200 fee for paperwork processing, materials, and other administrative costs associated with my care and they do not bill that to insurance. That seems bizarre to me - all administrative work is wrapped into the total cost of care from what I understand, which is billable to insurance.

Is this type of fee common at all birthing centers or is this facility specific? Has anyone else had to pay something like this? If I can get coverage at another birth center that doesn't charge the fee is it worth shopping around a little more?