If you've had a fraught relationship with your psychiatrist, I can see where that will feel invasive. For me there was no bar at all to getting hrt however I've been in therapy for gender identity (along with depression) for 20 years so my PCP didn't question that. Some docs may require a letter.
That said, your gender therapist is probably going to want to talk to your psychiatrist for any letter of referral - the hurdles for SRS are bigger, I don't know if that's relevant for you. For that to happen, yes you would need to give a release. You and your psychiatrist don't need to agree on definitions for you to receive HRT and as far as I know estrogen and testosterone hrt aren't contraindicated with any drug therapies.
WPATH guides what physicians must do to treat your gd/gid and your psychiatrist is more or less obligated to work towards your best health. If you refuse to allow the consultation I would be concerned you may have to engage in more extensive therapy to proceed with transition, especially if that eventually means GRS.
If I were in your shoes, I'd consider the following .... for GRS you would eventually need 2 letters, one must be from a PhD psychologist (I think an MD psychiatrist also works for that). If your other issues may be eased by HRT then your first doc has no reason to stand in the way. On the other hand I know I've seen a pshrink who had a different view of being female than I did and I think that was partly cis-gender assumptions on the other hand transition wasn't on the table then so it didn't really matter. My present therapist/psychiatrist isn't all that well versed in gender issues and wanted to consult with an expert. What came out of that was she now fully understands my gid and just yesterday has agreed on writing a letter for GRS.
If your psychiatrist really can't be anything but a roadblock then you may ultimately need to see a 3rd shrink. Meanwhile I'd say talk openly with both of them and see if you can get comfortable with the release.
I know that's vague, perhaps you can elaborate more on your difference with psychiatrist on gd?
Seeing the post made while I was writing, on the one hand I agree that psychiatrists sometimes know less than psychologists, happily mine while not specifically experienced has learned quickly and personally I love that as an MD she understands when I talk about biochemistry and that's been a great support as we've been able to get me past crippling depression.