I would say to search for a psychologist with a senior position (in my country someone with a senior NHS post) who specialises in the things you need help with. I know you said you don't have a choice of who to see because it's in the public health system. I guess then if you do find someone who is a specialist in the things you need, you could a) approach them to find out if there is any chance they can see you. For example, do they do any private work pro bono. Or would they be prepared to see you once for an assessment and then consult with your regular T, to give them some input or guidance. And/or b), you could take this issue to your regular T, and ask if they would be prepared to make contact with the specialist to see if they could see you both to give some input, or perhaps advise your regular T.

I know that some people have very good Ts who are not necessarily in senior positions, btw. I'm just saying that's what I would do, and that was the approach I took when I found my brilliant T - except that I was fortunate enough to be able to pay privately.