The further evidence I sent with my appeal papers name that school directly.
Applicants can only be ranked on the information available at the time of the application. Where a school has an Exceptional Medical / Social Need criterion, this is normally clarified in the admissions policy as requiring evidence from a relevant professional that the specific school is the only one which can meet that need.
So if the supporting documents submitted back in the autumn don't specify that, this 'evidence of need' is unlikely to have been sufficient.
Papers you are submitting now are being submitted to the Independent Appeal Panel for the purposes of their consideration of the strength of your case versus the school's case not to be able to admit any further pupils to the year group, not to the school for a sort of second go round of the coordinated admissions process. (Even if the person handling the administration of appeals is contacted via the school).
Moving on to your appeal, along with considering the question of Was there an error when this application was made, which cost this DC a place at the school?, the Independent Appeal Panel will look at all your reasons for wanting a place there, whether or not they relate directly to the oversubscription criteria. And will then decide whether your case is strong enough to overcome the prejudice caused to the school (its having to cope with an extra 'body' and yes, things like the stretching of finite resources such as support for pupils with additional needs but without an EHCP).
So the documents you can now provide with a recommendation that this is the school that your DC needs to attend are relevant to the appeal panel's consideration of your case, but even if they meet the criteria for demonstrating Exceptional Need, the school could not have considered them back when applications were ranked, because they didn't exist then to be considered, iyswim.