Apparently historians are agreed that the issue of Arthur's birthplace has been set to rest by the discovery of the Arthur family Bible. If there were such a thing as the Obama family Bible, I don't think it would suffice.
Actually, while it might not be sufficient, it might be an element. It's possible in Hawaii and other states to file for a delayed birth certificate, and for non delayed one that alleges a home birth. An entry in the family bible might support or refute the fact of birth at the time and place alleged in the application and accepted birth certificate. But in order to know if the Certification of Live Birth is based on such a "nonstandard" Certificate of Live Birth, one has to see the Certificate, since such information is not on the Certification. Filing as a home birth is just the sort of thing a concerned grandmother might do to ensure her grandchild was a counted as a US Citizen. Something most, even extreme liberals, thought important in 1961. But since the Certification released is a clear forgery, the Certificate could say almost anything.
The State Department will not accept Delayed Birth Certificates, ones that show a date of birth more than a year before the date accepted by the registrar. That's because such are so easy to get for persons born outside the country, and the Certification (or other abstract, different states have different terminology) does not contain enough information to verify that the birth actually took place where and when indicated.