Well, that's the key, isn't it?
Absent torches and pitchforks, the Court does nothing or at most opines and defers to an impeachment process that doesn't happen. But should torches and pitchforks materialize for real, the Congress impeaches and removes, so the Court needn't do anything. Either way, the Court is at most an adviser, not a remover.
The key is to get up the torches and pitchforks. Counter-factual (not born in Hawaii) or esoteric (all the Vattel stuff) arguments about eligibility won't do the job. What Obama is actually doing in office just might.
You can't impeach a usurper, and you don't need to. Maybe a "sense of Congress" resolution? But impeachment is for Presidents who have gone astray, not someone never eligible in the first place, and thus fraudulently occupying the office of President.