http://decalogosintl.org/documents/sos/Response_to_Discovery_Motion.pdf (pg 8 of 9)
Title 19. Health
~ Chapter 338. Vital Statistics
~~ Part I. State Public Health Statistics Act
~~~ § 338-18. Disclosure of records ...
(g) The department shall not issue a verification in lieu of a certified copy of any such record, or any part thereof, unless it is satisfied that the applicant requesting a verification is: ...
(4): “A private or government attorney who seeks to confirm information about a vital event relating to any such records which was acquired during the course of or for purposes of legal proceedings”
In theory, if an attorney has been responded to with a copy of the "CertificATION of Live Birth" from Camp Obama, it should give that attorney justification to request "verification in lieu of a certified copy."
You are assuming that the Obama camp has responded with a Certification of Live Birth. To my knowledge they have only responded with arguments against providing anything.
In Theory, a certified (paper) copy with a raised seal and registrar's signature on the back, would be sufficient to prove birth in Hawaii, or elsewhere. Even a "Late Registration" Would, unless the original information was falsified, show the correct location for the birth. However because of the relative ease of such falsification, a Certification based on such a late registration will not be accepted, such as for a US passport, without additional documentation, whereas a long form certificate might be.