I was going to say supporting and defending the constitution by oath of office, but nufsed beat me to it.
But that's not an answer. The constitution doesn't say he has to produce a birth certificate. It only says he has to be certain things. He can be those things, and still not produce a birth certificate.
So if you want to take someone into court for not following the law, that's not going to work. You have to be able to point to a law that says they must do a specific thing that they didn't do.
Come on, let's deal with this one concept without evasion. There's nothing wrong with acknowleding the reality.