The letters of support could definitely help, but could also mean exposing your personal life to people at your job/school just to get them to understand why you'd need the letter.
I got mine done though insurance from my job, and they had the same criteria (consistent with WPATH standards of care). Basically, I proved RLE within the letter from my mental health professional which was also mandatory for my insurance to cover it. He just sort of documented how long I had been presenting as male (clothes/passing), and what I had people call me. Luckily for me, my nickname became my legal name during transition so it was more convincing in terms of how long I had been going by my male name.
Long story short, you should have to present enough information to your shrink for he/she to attest to your RLE in the mental health screening letter. It was the least invasive and most private way to prove myself (bleh). I even had pictures of me growing up just in case.
Best of luck