Is the patient obligated to pay the doctor's fees outside the scope of the agreement, or is he considered not responsible for what transpires between the institution, the doctor, and the hospital?

You have no relation to what will transpire between the institution and the female doctor. The transaction is between you and the hospital and your workplace. The fee is paid by the hospital or the institution. There is no blame upon you for working with what the female doctor herself stated, that you have no relation to her fees.