Hi Determined Mum
Ok, so the Head wouldn't support at first, but then decided you should appeal once she found out who your son was?
(Sorry, a bit cynical - it's been a long hard day here!!)
It is NOT
too late for her to change her ranking for your son. The OoS may be engraved in stone, but not his ranking.
Go back to her and say you have it on good advice that she can state that she was not aware of the tremendous progress he has made in the last few months. On reviewing his case she is now aware that her ranking for this particular child was too negative and based on discussions with his form teacher, she would now rank him as X:X, which would place him approximately Xth on the OoS.
If the school has a lot of GS passes and her previous track record on rankings has been good (ask her how accurate it was - how many of he top 10 actually passed? and the top 20? and how many surprises at the bottom?) then the panel will accept her new ranking for him. You could even ask for a copy of the OoS for the school, with the pupils' names blanked out - you will receive that anyway in the Appeals information, so she would just be giving it to you early.
The 11+ scores are still on the low side, so your appeal is still rather uncertain. However, if you can give the panel a convincing argument that much of his progress has been achieved within recent months and back that up with evidence from the Head, form teacher and any other relevant teachers, who knows?
I think you need to go back and politely test out just whether the Head is really willing to support you with this new information.
If all else fails, look at the Bucks section for information on the 12+, as I would suggest all near-miss 11+ candidates consider it.