It is difficult to know whether to advise you to move house! The chance of a successful appeal at 11+ is around 40%, less for a lower score. It will all depend on the usual criteria - demonstrable academic achievement, any mitigating circumstances, strength of support from the school, etc.
If the Head will not support you because of "political reasons" then state that - it should not count against you. However, it will mean that the rest of your case should be as watertight as possible. If your son had good scores in the official practice papers then that could be very helpful.
There is a 12+, and also 13+ exam in Bucks. Until a couple of years ago a number of places were reserved at grammar schools for children taking the "Late Transfer" tests, as they are known. However that was discontinued, and now the children who pass are only given a place if a vacancy is available at their chosen school. Any vacant places are allocated on the standard Bucks criteria, of which the first is distance from school gates. If your preference is for a very heavily over-subscribed school there is almost no chance of gaining a place unless you are living very close to the school.
The right to sit the 12+/13+ is not automatic - you must have the support of the Head, and the child should have achieved at least 2xLevel 5 SATs. (Note that this requirement is not published, but is generally known to be the base criterion used by Bucks.)
The full process can be found on the Bucks CC website:
As you will see, at 12+ there is a reasonable chance - around 70% - of the child getting into one of their preferred schools (not necessarily their first choice school). At 13+ the chances are below 40%.
The one word of caution I would give is that, if your appeal doesn't succeed and he doesn't pass the 12+, the alternative Upper Schools are not always great. You need to look very carefully at what the fallback option would be.
I hope that helps you in some way.