I think contesting the place of birth is also a good idea because there is enough stuff floating around to make the place of birth also questionable. I don't think he has shown anyone his "original" birth certificate. I think what he has put forth is a Hawaiian DOH created replacement birth certificate issued to Adopted children, and designed to LOOK like an original.
Hawaii has peculiar birth certificate laws. You don't actually have to be born there to get a Hawaiian birth certificate.
It is more risky for our side. I'm pretty sure the court is going to uphold Wong Kim Ark and decide that it is the relevant precedent that matters. If they ruled the other way, it would overturn the citizenship of millions of anchor babies.
In order for them to overturn anchor babies, they would have to rule Amendment XIV did not include children born in the United States to foreign parents.
Since Amendment XIV is irrelevant to Natural Born Citizenship, it does not require any explanation thereof. They simply rule:
Natural Born Citizens are born in the United States to parents who both are, at the time of birth, Citizens of the United States.
That does in no way affect anything else in the Constitution. Problem solved.