Leslie Thompkins should probably be Martha's OB. As a close friend to Thomas, she could be trusted to keep Martha's miscarriage a secret, and also to falsify records for Bruce's "birth". She could also serve as a surrogate mother to Bruce, as Alfred acts as his father. This way, only the two of them will know Bruce's secret.
I'm not sure when his powers should manifest yet. The ramifications for different periods would be significant.
This would allow him to grow up knowing of his powers and allow his parents to guide him in their proper use. This also means that he could have prevented their deaths, and failed to (likely because he was scared by Joe Chill). Along with a need for vengeance, he'd carry some significant guilt with him.
This might allow some amount of Superboy stuff in childhood.
When Chill attacks him, Bruce would still be unsure of his powers. This would still leave him with a sense of guilt, but would better explain his inaction. This wouldn't allow for any Superboy activity, but might still allow for some during teen years.
Perhaps, he could use his powers irresponsibly in this scenario as a teen. Might also lead to a visit by the Legion of Super Heroes.
Chill could try to shoot him as well, but the bullet could be deflected by his invulnerable skin. This would be the first time his powers manifest.
This would be harder to do, as he would be dealing with manifesting powers and parents' deaths at the same time. This would put a significant burden on Alfred.
In this case, Alfred would be the one to explain about his extraterrestrial origins. This would be similar to option 3, but would add to his anger that his powers hadn't manifested in time to save his parents. He might be even more likely to use his power irresponsibly as a teenager once they manifest, during a time when he doesn't yet have full control.