I agree with your point that he should have presented the documentation before it ever came to a lawsuit.
But his representation should have known that in Quo Warranto he could be required to show the documentation anyway so it would save a lot of trouble just to do it right away anyway.
And (though I know this wasn’t in your point at all; you made your point well) if the homeschooler presented fake documents and posted them online and had people make public statements about it, it would be more analogous.
My secretary of state told me that there is no legal requirement for the documents to be produced because the people have the power to vet the candidates. If Obama had merely made the argument that he doesn’t legally have to produce the documents the public would have possibly said, “Then screw you.” What he did was say he didn’t have to produce the legal documents but then produced illegal ones. And Americans swallowed it.
DOJ argued to Judge Carter that quo warranto doesn't apply to POTUS so they intend to fight against any quo warranto.
Judge Carter didn't exactly agree or disagree, but he did state that quo warranto against a federal officer may be brought only in the DC Circuit.