it does come down to that. doctor/patient privilege is one thing. but that mostly applies to court/legal issues. when it comes to insurance issues, he is pretty much duty bound to tell them all
there are two options. one, see a DR "off record". iow, one that is not paid for through your insurance. this MIGHT be worth it if he is very progressive and willing to work with you. certainly, many HRT docs would fit in this category
two. more problematic. see if you can talk to your DR, but "off record" or "hypothetical".
if you get his agreement beforehand that what you are going to discuss will be off your medical record/chart and/or "hypothetical", then he might agree to speak with you this way. his #1 concern is theoretically your health, not being a secretary for insurance.