The law that Congress passed to “prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof” said that the way to prove that it is truly the record of the state is with a seal and signature. It’s basically saying that a certified record has to be acknowledged as a record of that state.
That doesn’t mean that another state can’t require more proof than just that record.
For instance, if I used a Nebraska COLB to try to claim that I qualify for DHHL land in Hawaii, Hawaii would not consider my Nebraska COLB as sufficient proof (just as they don’t consider a Hawaii COLB as sufficient proof).
Is that constitutional, or does it violate the Full Faith and Credit Clause? Why or why not?
You're right, the Constitution doesn't say anything about birth certificates and so a state could presumably come up with some different kind of test. Maybe a state could require that a candidate produce a live witness who can testify that he/she was present at the birth and observed the child often enough thereafter that he/she is certain that the candidate is the same person he/she saw being born. But, who can pass that test?
I don't think the Full Faith & Credit clause requires the use of birth certificates, but if they are used, I believe that a state will have to accept the certificate recognized by the issuing state as a public record.
Wouldn't it be easier for the House of Representatives to just subpoena the Hawaiian records?