I would absolutely tell your pdoc that you don't want your primary doc involved in your mental health management. Even if you can't explain why, just saying "no" should be enough. I think it would be helpful and validating to explain to your new pdoc why you don't want primary involved, but you don't have to.

I recently told my pdoc that if he needs me to have any blood work done because of my new meds, I will NOT be going to my primary doc. He said that's not a problem, he can write a prescription for the tests I need and I can take it to any lab that does blood work.

Personally, I think I'd be looking at getting a new primary care doctor. There are ones out there that are much more sensitive to mental heath issues and the vulnerability one feels during exams. It took me some trial and error, but I found both a primary care doc and an ob/gyn who are both phenomenal. Both do their exams and are gentle and understanding and talk me through every step. Then, as soon as they're done, they instantly step out, let me get dressed, and then come back in to discuss anything they need to. It did take me not settling for the first doctors I met and it did take me insisting on consultations and talking to the doctors in their offices before the first exam to find the right doctors for me, but it is possible.