So I would say that the Tavi position is tenable, but only if they present their own gender views as belief rather than fact
Well, yes - it would be a reasonable position if the point of their organisation was to be a Church of Gender.
But they're supposed to be a medical service, part of the NHS, working with people who are confronting their desire to be the opposite sex when they cannot actually change that.
Insisting people deny reality relevant to their field of operation clearly does not serve their purpose in the way necessary for such discrimination to be permitted under the EA2010.
Best you can argue is that the "best treatment" for such individuals is for everyone involved, even tangentially, as overseers, to participate in the lie. I don't think that would stand up though - the person overseeing a role play has to know it's just a role play to run it effectively for its participants.
(I recommend the film Synecdoche, New York btw, which is on this theme - a playwright + stage director gets a bit too caught up in his own play.)
But if it is a belief-based organisation like a Church, rather than an evidence-based organisation, I think they'd be okay? They'd need to change their terms of operation though, and surely they would not be eligible for NHS funding?