I am sorry to hear about all that.
Firstly, the deadline for submitting a Bucks appeal has now passed, so unless you have submitted the application form, I am afraid that you will face an almighty struggle to get an Appeal hearing now.
On a score of 107 (and perhaps a lower "other" score) and without the support of the Head you would, I am afraid, have very little chance of success - less than 10%. You would need absolutely sparkling school reports to get through without his support, and very significant mitigating circumstances.
The mitigating circumstances are mixed - the loss of his grandfather might have explained the shortfall in marks - say 117/118, but you would really need independent evidence from the school or a doctor of the effect it had on him.
The anti-biotics for tonsilitis would not make a good point, because Bucks give the advice that you should not send a sick child in to school to take the test. Even if he was recovered from the actual illness, you would still find it difficult to convince a panel on this point.
It is particularly frustrating that your son appears to have taken the "equating test" and passed with a score of 126. (The equating test is the first paper from last year, and is only done by a handful of schools in Bucks. The purpose of it is to evaluate the relative difficulty of this year's papers vs. last year's. A complex business!)
I am not sure from your post whether the Head's recommendation was a 2:2 or a 2:4. If it was the latter I suggest that you write to the Head in January, stating that you will not be appealing, but asking him to clarify why his recommendation was so low. With predicted 3 x 5a SATs it does seem pretty extraordinary.
I suspect however that the Head is not willing to support your appeal simply because the score is such a long way from the pass mark. The cynic in me also wonders if this is a Prep School with a Year 7 intake?
I am sorry that I can't offer you better news. I suggest that you take a look at my Sticky on Bucks about the 12+/13+.