No. At risk of starting the "backlash," there is no need to call him "she" and "her." There is, yes, a need to refer to the reproductive body parts.

But that's not an excuse for misgendering him. It might take some getting used to..."birth father" is not a phrase that's been used much yet...but if it's constant, then it shows a lack of care and consideration for the patient, which is a red flag in ANY care team.