Hypothetically, is it better to get the case dismissed for lack of jurisdiction, or produce a document that will put the matter to rest?
Would one document put the matter to rest? The plaintiffs would demand that forensic document analysts verify the authenticity of the document. They would also demand discovery into Obama's passport files, his college records, his parents' divorce case, and all sorts of things. (Read some of Orly Taitz's court pleadings.) Finally, they would say that even if he was born in Hawaii, Obama still isn't a Natural Born Citizen because both his parents weren't citizens. (This has been alleged in many of the court cases.) The result would be that the court cases would drag on for 4 years or more, all through which Obama's opponents would claim that nothing he did was legitimate because the court hadn't ruled yet.
If I were Obama's lawyer (and I'm not!), I would be doing exactly what he is doing.