So what you're saying is that you had evidence to support you that you met the exceptional medical and social category criteria due to "x, and y", and applied on those grounds, but were refused on the grounds that:
a) you didn't have an EHCP
And
b) they couldn't meet need "z", which your child neither has, nor does any of your supporting evidence mention.
Is that right?
So if an appeal panel was convinced that there was no need to have an EHCP to meet the social and medical need criteria (which is the whole point of having one, because an EHCP skips all criteria and goes to the top of the queue), and your child doesn't have need "z", which erases the school's problem with meeting need, then they should, in fact, be able to take the child, which would just leave the issue of whether the evidence you have is of sufficient weight to argue that only that school can take your child.
Do the letters say that in the writer's opinion this was only school that would meet the need of your child, or do they say that 'user1488391504 believes this school is best for her child' or some such phrasing?