How on earth have you managed to avoid a GIC for that long?! :-) lol. Nottingham are so stupid about how long people stay in their system. Some people I know have been through in one visit with that much time. They made me wait the whole time too and I had already changed my passport, name, was FT for months and been on HRT for a fair while. Anywhooz...
If you wait for a diagnosis of gender dysphoria from the GIC (what a joke with that much time under your belt) then you will probably wait for three months beyond when your next appointment is. that gives you 3-6 months before you would be able to ask them, unless they give in and diagnose the next visit.
If you think you can get letters from suitable people in the meantime then go for it, but I think it will probably take you that long or longer to find suitable people to write to support you and get letters from them. If the PhD requirement is accurate and I have no reason to believe otherwise, then Dr. Curtis wouldn't do. He is a GP type Dr and their 'Dr' is not from doctoral level study and award, its an awarded title only, and their actual qualification level is to graduate degree (though they have two of them). His highest qualification is a Masters level in Business as I understand it.
If it were me, then I would let them know that you expect letters from them on your third visit, when they make the diagnosis. Make sure you can prove the 30 months and RLE time with photos so that they can clearly see that you meet the requirements for the letter. If they do diagnose on the second visit, ask them to forward a letter to you and cc your GP.
That's what I would do.
Good luck x