Last things first - at this stage, the school is making a case to the independent appeal panel as to why, with ot already being at PAN (or above) for the relevant year group, it cannot admit any further pupil(s). There should be nothing personal regarding your own (or any other appellant’s) DC, other than confirmation as to why your DS was not ranked within the number who could not be offered a place when all applicants were ranked. I note that you say you have been told 'distance', so I would assume that this area is overwhelmed with boys of the RC faith? It's somewhat different where I am, where the second nearest Catholic comprehensive usually goes down to 'all other applicants' albeit occasionally only 'with a sibling at the school).
The 'medical' supporting evidence for why you want a place at the school would only have been relevant if the school actually has a relevant oversubscription criterion and
- whoever was responsible for scrutinising the evidence deemed it to be strong enough for your DS to be ranked under that criterion and
- that a ranking under that criterion wod have been high enough for a place to be offered.
However, even if the school has no 'medical / social need' oversubscription criterion, if your DS's needs in this area are the reason why you believe that this is the best school for him, then the documentation should certainly be submitted to the clerk to the appeal panel as part of your case. You should have been given a deadline for doing this to guarantee that the panel will consider it - please don't just wave it under their noses on the day of the appeal, as the Chair can refuse to accept paperwork at that stage.
It will help your case if the professionals involved have stated that in their own opinion, based on their knowledge of your DS, this school is best placed to meet his needs, and why. Not just 'Mum tells me that...'
What conversations have you, or the SENDCo at your DS's primary school had with each of the schools concerned about how they provide support for pupils flagged up.as potenthaving more problems than the average with the transition from primary to secondary?