THIS OBVIOUSLY NEEDS TO BE VETTED...
... could this match where he was born -- maybe all of the databases hadn't been scrubbed...
Here's a variation of the theories that would match:
After mother Ann Dunham realized her marriage in Kenya wasn't working (mentioned in many places), she flew West instead of East to give birth to Barack Jr in Massachusetts in early to mid-summer 61. She stayed with a friend of the family on her or Barack Sr's side near Cambridge, where Barack Sr would attend the following year.
Several weeks or months passed, and being new in town, she realized 1960's Massachusetts was not very accepting of a mixed-race child where the father wasn't around. But Hawaii has always been a crossroads of ethnicities from different countries and continents -- Barack Jr would be more easily accepted.
Ann wrote or called her mom, Madelyn Dunham, who made arrangements for Ann's and baby Barack's arrival in a week or so. Toot got all of the necessary forms to help her daughter register her grandson in Hawaii with a "Certificate of Hawaiian Birth" or "Certificate of Delayed Birth." Newspaper policy was pretty "hang loose," so the clerk accepted the small fee to publish a birth announcement on Aug 8, 1961 to help with the evidential evidence for the paperwork, perhaps claiming it was a home birth. The unsuspecting newspaper clerk, accepts the fee and info directly from the grandmother, considering Barack's birth to be a "home" birth.
When Ann get in Hawaii, she can begin to register Barack Jr's birth in Hawaii via a "Certificate of Hawaiian Birth" or a "Certificate of Delayed Birth." His birthday will be registered as 08-04-61, but he had actually been born several weeks earlier back in Massachusetts, perhaps even passing through Seattle on the way to her new life in Hawaii. Once the baby birth announcement is printed in the local newspaper -- just sign some affidavits by an assisting-Toot, and submit the affidavits and newspaper birth announcement to turn in for a the "Certificate of Hawaiian Birth." Wait a few more months and convert the "Certificate of Hawaiian Birth" into a "Certificate of Late Birth" -- just provide a little more details on the supplementary affidavits. ALL like Hawaiian Law ALLOWS then, and NOW.
Perhaps that's why no one's found a "birth certificate" in any of the hospitals in Hawaii ... or Kenya.