To be honest, I really don’t know what all this terminology ultimately means.
What do you need to get a passport? Whatever that is, they would enter that.
And from your perspective, why does the birth certificate matter? You know he has a parent who was an alien.
If you are hung up on this two citizen parents nonsense, why do you care about the birth certificate? You should be trying to get his memoirs entered as evidence.
Following on that thought...If I were to be asked by someone (say, the DMV), for a copy of my b.c., and I gave them a URL for where they could view an image of my b.c., there is a zero percent chance that they would accept that on "face value."
The birth certificate is secondary, but still important because as we've learned, it was legal for a baby born anywhere in the world to be eligible to apply for a HI birth certificate. It would help to establish weather or not he's a "citizen."
The bottom line remains, born in HI or born somewhere else, he was born a subject of the British crown. And that, is no nonsense.