They would have to. I imagine it would be immediately appealed or else the military would be dealing with legions of troops trying the same thing. That would completely disrupt the military and give a "get out of jail free" card to anyone who doesn't want to deploy --whether they cared about Obama's eligibility or not.
Should such a verdict happen, and I very strongly doubt it will, you can guarantee the issue of Obama's eligibility would be thrust to the headline story of every news media imaginable.
Or we fix the problem by getting an eligible President. Even if that's Joe Biden.