In 1961 the birth of a "black" baby to a white mother named Stanley would be sufficiently unusual to stick in most people's minds. They might have brought that memory out to "look at it", until something, say the canidacy of BHO. a "black" man with a white mother born in Hawaii, would trigger the memory.
From my own experience, I doubt it. The doc would have to be aware that the baby was half black as opposed to some other type of mixed race child, which he wouldn't necessarily realize. (Mixed race children are not rare in Hawaii.) As I've said, I could have delivered someone famous, and I wouldn't know it. I don't recall details of the appearance of women that I delivered, and usually didn't see details on the birth certificate that would have told me the race or the name.
You're asking a lot of hard working nurses and doctors - without an ongoing relationship with mother and/or child, it's extremely unlikely. I think there are stronger arguments. This one will get dismissed by medical people know from experience.