Slightly confused as to not knowing he had medical needs prior, usually there'd be signs if it was serious enough to warrant not wearing a mask (including trauma/sensitivity to things). I wouldn't mention a solicitor, as I think approaching schools on a 'chat' level with respect is better, otherwise they'd feel threatened. I had an incident in my son's class (Covid, but not mask related) and just sent an email saying could they address it with the class (didn't want to mention names/single out), when they replied I thanked them. No need for a solicitor.
Personally, I won't be sending a letter like that as my child isn't exempt and I want them to wear a mask to help out. I also believe in building resilience by helping them appreciate sometimes we do things for ourselves or others that we don't like or that feel strange, but we can get through that. You don't need a 'condition' to be exempt partly due to the difficulty in accessing a diagnosis for problems like PTSD/C-PTSD for sexual abuse victims and the fact that the child/adult may not have sought help for this or have people around them aware. They also may not have had the appointment for diagnosis, been observed for long enough or may have a diagnosis that doesn't acknowledge trauma. They (or their parents) may not want or feel it needed for the school to know they went through that. Equally, there are many other reasons where an issue may not neatly fit a condition or diagnosis may not be needed (eg. working with people who lip read).
However, if this is about discomfort or becoming upset because he doesn't want to wear it, then that's a bit precious, maybe? It's important to wear the mask, this rule is sensible and protects ourselves, to a lesser extent, and others. There will be vulnerable children (and probably adults) exposed to your son, who isn't wearing a mask. While this is perfectly acceptable if he has real issues that can be identified (whether they fit a diagnostic criteria neatly or not), it isn't acceptable if he doesn't. It seems unusual to reach teen years (assumed as mask is issue, not so much for younger children) and not be aware of hypersensitivity issues, trauma (possible), asthma/breathing difficulties etc.
Obviously, you don't need to clarify what the issue is here, but if it's some anxiety/distress (not relating to trauma) or discomfort, I'd suggest encouraging him to wear a mask and finding one he can wear. There is no exact exemptions because they will inevitably miss some out, however, you need to ask if your son was stopped by Police would he be able to explain the exemption (or if he has an issue that affects this, could an appropriate adult)? I find that a useful boundary - same for the 'except in emergencies or to provide care' part of the full lockdown rules.
It's not really a legal dilemma, but a moral one. If you are certain your son's reason is good enough, then by all means let the school know. I would do away with the solicitor and other people won't need encouraging - most people will be aware if their child cannot wear a mask and will have let the school know in their own way. If their child should not be exempt (as in our case), they shouldn't be sending the letter.
My letter would be more along the lines of... 'Dear X, I just wanted to let you know that DS is unable to wear a mask due to [reason]. Could you let the teachers know if possible? I'll get him an exemption card to help avoid confusion. He has been reminded that social distancing and good hygiene is very important at the moment. Kind regards, Mrs. Magpie'