He's got a Hawaii birth certificate, he just doesn't want to show it.
There are a number of reasons why this could be so. The actual certificate could show that his father is somebody other than Barack Obama, or just "father unknown" or "illegitimate". That would explode his entire self-manufactured "biography" - he just can't afford to make it public.
I'm inclined to think that his mama, being a deep-dyed Communist and free-love advocate, had no idea who the daddy was. And her parents, being not Red but merely Parlor Pinks, set about trying to give the poor little baby an at least arguable father, by planting all the information they possibly could. Hence selecting as the putative father an exchange student who was going back home in a few months. Hence the newspaper announcements giving a residence where neither Barack Obama nor Stanley Ann had ever lived - the confusion over the hospitals (to prevent anybody from locating the actual certificate) - holding the grandmother incommunicado - and so forth.
It all smacks of a valiant attempt by grandparents with some remnants of middle class propriety trying to save their grandchild from the stigma of bastardy. And in the 1960s, even in Hawaii, there was definitely a stigma. People in this day of casual divorce and illegitimacy don't realize just how bad it was to not know who your daddy was, and what sort of person that made your mama.
Didn’t the STATE law in Hawaii at the time suggest that children could be born ‘anywhere’, as long as they got to the Big Island within a year to register the birth and apply for a certificate? Like there were other islands and it was difficult to get to the Capitol to register a child, so they gave them a year to come and register a birth?