It might just be an internal policy of theirs - wherein, if they were to provide you with a letter to update your gender marker, they feel obligated to update your gender marker in their records which could then lead to issues with insurance claims. They want to get paid, so they don't want to do something that would cause their money to get held up. You could try telling them that you need a letter from the doctor to change it on your insurance?
It's literally just a form letter though for a Passport. They don't have to write anything, just sign it and put the medical license number. Not sure why it'd be a big deal.