we feel that we have a case, I just don’t know how strong
I do believe you have a case. Whether it's strong enough, only your particular appeal panel can say for sure. One might expect panels for superselectives to be quite tough - but I would hope they would note the 'very superior' results from the EP, especially in the relevant area of perceptual reasoning.
We have wondered whether he is dyspraxic and as early as the first term at school his poor motor skills were noticed and he was included in a “Jump Ahead” programme to try to improve them.
It helps that you have evidence of the motor skills problem. Did the EP not comment on dyspraxia?
The main area it affects him (apart from coming last in every race on sports day!) is his handwriting, which is appalling and slow. I am worried about taking his exercise books to an appeal as they look so terrible, full of comments about poor presentation.
But what about achievement and the underlying quality of the work? The presentation simply proves your point, and shouldn't matter so long as he's consistently scoring very high marks.
He has struggled with friendships at primary school; he knows so much about things and is quite intense. The other children don’t dislike him, they just don’t really “get” him. We thought that if he was in a selective environment then the “cleverness” issue would be taken out of the equation socially (yes, I’m sorry, I’m aware how awful that sounds!). He has also been bullied this year.
It would help to have evidence from the primary school to cover these points.
Note that they might be an argument for a selective school, but they're not in themselves an argument for a particular school. However, if his 'one and only friend' were to be going to Judd, then a relevant argument would start to take shape.
So which angle to we use for appeal: poor exam technique on the day but otherwise high academic achievement? Poor motor skills and slow processing speed? Emotional well-being? Do we even mention all of this?
All - but, ideally, emotional well-being needs to be linked specifically to the school in question.