I agree it would be an unmitigated catastrophe. Personally I think it would suck bad to have McCain win on a technicality (no matter how important the technicality).
But in this case, he has nothing to win by cooperating immediately.
This way he can keep the Rep Mudslingers occupied.
If McCain should openly attack, he can just say 'Oh here are my papers' and McCain will appear childish and lose more points. Worst comes to worst, there will be a court case, he will show his papers and that'll be it. And he will still be able to put a negative spin on this for McCain.
Perhaps you are someone who cares, but 99.99% of the people will assume that he is American unless he is proven not to.