what else can I do?
This white coat syndrome has plagued me throughout my career. I have had hospital stays and, after the initial tension, blood pressure goes to normal. Like you, I monitor it at home and it's always in the normal range. I've worked out my entire adult life and try to keep the weight down with a (mostly) healthy diet. Airline work is sedentary so it's important to get to the gym regularly. I refuse to be diagnosed or medicated for hypertension based on the relatively momentary nature of white coat. And I hate someone telling me to "relax." It's like telling someone to absolutely not think about oranges. If the livelihoods of these physicians and nurses depended on a physical every six months they might be as nervous as you and me.
So, back to your question.
If you can, find an AME who understands. The best way I've found is word of mouth among airline pilots. When I displaced to a new location, I asked around. I also looked for somewhat out of shape guys to find out who they went to. I received a lot of wry smiles and some great information. The first AME I went to when I was new to the industry was recommended to me by a somewhat stout guy and I went to him for years. He understood. When he finally retired I went to the guy who replaced him who didn't understand. Because I had names of alternate AMEs I found one who I continue to go to. He understands. Good and experienced AMEs understand and pilots continue to go back to doctors like this. They understand white coat and that it doesn't constitute a threat to your own health or to the national airspace system. They have the big picture. No pilot wants to go to a "hard ass" AME. When an AME gets a bad reputation, pilot business likely suffers.
As an aside, linking your GP and AME together can be hazardous to your livelihood. Go to your GP for a health issue, go to your AME for your livelihood. May the twain never meet. I know, that's not always possible.
Best of luck on what I know is a worry and frustration. I walk in your shoes.