Since there is no known record of Barry renouncing his British citizenship and no British legal "act" that would have removed his British citizenship (obtained AT BIRTH)...he continues to be a British citizen today and perhaps a U.S. Citizen (assuming birth in HI...yet to be proven).
So, at this point, "best case" scenario for him is he is a dual citizen (British & U.S.). Thus the question, of weather or not International Law (& U.S. Law) recognizes a dual citizen's ability to enter into certain (all?) treaties.
It’s not so much that his father was a British subject, and thus he is a British subject under *British* law, but rather that his father was *NOT* a US Citizen, that is problematic. US law, common or statute, doesn’t care what British law, especially statute law, has to say.