I'm sorry to hear about the difficult time you have been through.
What "reservations" did the Head mention? Are you in Bucks? If so, what ranking did the Head give Alex?
My suggestion is that you must focus on two separate issues.
Firstly, Alex had a very positive 11+ result at 120 (what was his score for the other paper, & do you have any previous VR/NVR test results from school?) and he is very close indeed to the required level of suitability for GS. Bucks claim that the 11+ is a test of raw ability, not current academic achievement, so is very worthy of consideration based on that result.
Secondly, the predicted SATs of Level 4 are (using Bucks CC's own logic) a separate issue, and reflect his current level of academic achievement. This is where the extentuating circumstances, supported by the Head, really come into play. You can argue, with the Head's support, that his predicted SATs do not reflect his ability but his response to learning over the past 3 years.
As his performance had improved since July the extenuating circumstances are not going to have a direct bearing on his 11+ result in the eyes of the panel (unlike, for example, a family crisis during the immediate period of the 11+). However, you could perhaps add that as his academic focus had been depressed by your home situation, his vocabulary - an important element of the 11+ - was not as well-developed as it could have been in normal circumstances.
I would not tell the panel of your views on education - everyone going to appeal feels the same way, so that won't separate you from the common herd!
Please let us have more information so that we can help you further.