You are on the hook to pay

First, whether the procedure is effective or not is irrelavant to this case.

Second, the destist did you a favor by checking with the insurance company first. The estimate she gave you was only informational, not contractual. It's your responsibility to pay the bill (either by insurance company or by yourself).

Based on the above two points, you have no bases to win. If I were you, I would choose to settle (negotiate an amount that's acceptable to both of you and pay it).