Birth certificates, official ones, are kept by the States, not by hospitals. The state claims to have a copy of his birth certificate, but they won't/can't release it's contents without his permission or a court order. The BC would show which hospital, or at least which country, he was born in. (It was possible to file for a Hawaian BC if the parents were Hawaian residents, even if the baby was not born in Hawaii, but that would be reflected on the long form BC. It is, or was, also possible to "fake" a birth in Hawaii, or most any state, by claiming an unattended home birth, up to several months after the claimed birth date. The child need not even be resident in that state, he could still be wherever the birth actually occurred, as the form doesn't require foot prints or anything other than a couple of signatures. One of a parent (BHO Sr, was probably in Hawaii, and of course the mail was functioning) one of a witness to the birth. That's it.
It's pretty certain that the address given for BHO's parents was bogus, his father had an appartment in a different part of town closer to the University. No one has been able to come up with any phone book or directory that shows BHO Sr and/or Stanley Ann living at the address listed in the birth announcemnt printed in the local papers An announcement that doesn't even say what hospital the birth occured in, nor did any of the others printed that same day. My hometown (now nearly 200K people, then about 125K) paper still prints birth announcements, with the name of the hospital. They even print them for cases of unmarried parents, and some where only the mother's name is given. However I think they now have to get the parents permission to print them, which the hospital records folks do.
I wonder if the Punahou School would have a copy of his long form BC in their files.
Usually schools request a BC when a student is being enrolled.