From looking around the web, it does appear that the state of Hawaii provides a mechanism to verify information about a birth certificate without actually receiving a copy of the certificate:
(http://hawaii.gov/health/vital-records/vital-records/vital_records.html)
Letters of Verification
Letters of verification may be issued in lieu of certified copies (HRS §338-14.3). This document verifies the existence of a birth/death/marriage/divorce certificate on file with the Department of Health and any other information that the applicant provides to be verified relating to the vital event. (For example, that a certain named individual was born on a certain date at a certain place.) The verification process will not, however, disclose information about the vital event contained within the certificate that is unknown to and not provided by the applicant in the request.
Letters of verification are requested in similar fashion and using the same request forms as for certified copies.
The fee for a letter of verification is $5 per letter.
I would think that calling the Vital Records Office would be enough to get an explanation of how to write for verification that Barack Hussein Obama, Jr., was born in the state of Hawaii on August 4th, 1961, to Barack Hussein Obama, Sr., and Ann Dunham. (The web page says "Letters of verification are requested in similar fashion and using the same request forms as for certified copies", but I didn't see anywhere on the provided PDFs to request verification rather than a certified copy, so I think that it might take a telephone call to ensure that the verification request is being done correctly.)
The end product would hopefully be that one gets a "yes" or "no" answer to the question from the Vital Records department.
Well, at $5 a pop, we're not talking about a lot of money to ask all the questions that need to be asked to whittle this down.