1961 Hawaiian long form birth certificates have no field for "religion", nor does any other US one I've seen.
By US law, a parent cannot forefite the citizenship of a child. Nor can the child renounce it themselves, until they reach a more mature age, generally 18, but in some cases younger if they can convince a US consol that they are not being influenced or pressured by anyone to renounce their US citizenhip. And the bar for that is very high.
I would just point out that this a fairly recent interpretation of citizenship laws. In the laws in effect at the time Obama would have been adopted by Lolo Soetoro, there was nothing in the law that specifically prevented a minor from renouncing his own citizenship. It's also not clear what effect an outgoing adoption can have on a child's citizenship. One state department Web site acknowledged that the United States couldn't protect the citizenship rights of children adopted into non-Hague Resolution countries, such as Indonesia. What I think might be more overriding however is if Obama chose to exercise any of his Indonesian citizenship rights as an adult. It's pretty hard to claim NBC-status under such a scenario.