..... additional time but no percentages
I don't think this is necessarily a problem. Whatever the recommendation (often 25%), most children who are deemed eligible probably get much less in practice. The special access panel will make its own decision on how much extra time (if any).
He has managed to function well as he is bright and copes but for this reason he is ignored and not seen as struggling in my opinion
I fear this can often happen with brighter children who seem to be coping, and this will be a situation familiar to many appeal panels.
Now I'm wondering whether to submit the private EAP
I'd suggest going ahead, because I really don't feel you have anything to lose. You may well not get any
extra time, but you will have put down a 'marker' for appeal purposes.
Having said that, I suspect the main issue for any future appeal against non-qualification won't be extenuating circumstances such as dyspraxia, but whether you have enough alternative evidence of high academic ability.