Thanks for your email.
I'm not sure how much lower than expected your son's score was, but the further away from the required mark, the more need you have of extenuating circumstances.
The illness is unlikely to be enough (unless it was a 'near miss') because, as you recognise, you decided he was fit enough to attend, and don't have confirmation from the GP. I suggest you ask the school to state in writing the dates of his absence.
If you can prove that anything else was happening round about the time of the tests, or that school work was affected at that time, so much the better. With or without proof, I do think you may need to mention the other mitigating circumstances, so that there's a chance they will be taken into account.
I understand the point about not wishing to dilute your case. To avoid giving the impression that you are just coming up with as many 'excuses' as possible, you should be careful about how your statement of appeal is balanced. I would suggest at least a third of your statement should be on the academic side, no more than a third on extenuating circumstances, and at least a third on why this particular grammar school would be the best school for your son.
This doesn't apply to evidence. Gather together as much evidence as you can, and attach as separate appendices.
Hope this helps.