Separate the two problems here, because they need different fixes. The first is access to a test site. The second is whether 11 AP exams in one spring is a workable load, especially with two calculus based physics exams in the mix.
On access, get the answer in writing from the AP coordinator, not secondhand. Ask specifically: is the school registered as a test center, will it order exams for a course it does not offer, and what is the ordering deadline? If the answer is no, ask neighboring districts in the fall, not in the spring, and ask whether they accept outside students for AP exams at all. Some schools say no because of seating, not because of College Board rules, and that answer is easier to work around early. Also ask whether the two Physics C exams are even scheduled in a way that allows both in one year at that site, since Mechanics and Electricity and Magnetism are separate exams.
On the load, I would be blunt with your son about sequencing. Mechanics should come first, and it should be solid before Electricity and Magnetism gets serious time. Electricity and Magnetism leans on calculus and on the same habits, like setting up integrals for charge distributions and checking units, that Mechanics builds. If Mechanics practice scores are not consistently in the 4 to 5 range by February, adding the second exam usually just splits study time and lowers both scores.
A practical way to test this: have him do one timed Mechanics practice section now, under real timing, and score it by topic, not just by total. If the misses are mostly calculus setup, that is fixable with focused drills. If the misses are mostly concepts, like rotational motion or energy, he needs more teaching time, not more practice tests, and that is a reason to drop one exam or to take a dual enrollment physics course instead, if one is actually available.
Colleges will not hold it against him that his school does not offer Physics C. What they will notice is a transcript that shows he took the hardest physics his school did offer, plus a clear explanation in the school profile. Chasing two extra exams at the cost of grades in his other nine, plus two long trips, is a worse trade than one strong Physics C score and a sane May.