Well, depending on the particular elected office (and especially in state positions), the judiciary may have the power to review the election results and remove a fraudulently elected official even after his installation. But, that doesn't mean that the actions taken by that official prior to his removal would be nullified.
This particular case is problematic because the President has already been installed in office, and the judiciary has no constitutional authority to remove a sitting President, even if that installation was defective in some way. And, if they can't remove him, then they have no business examining his credentials ex post facto, or so they would argue. That's fine. That's why we have a Congress, who is Constitutionally vested to address such infirmities in the Executive via impeachment.
If Congress refuses to act, then the people have the power of the ballot box, and can replace those representatives with ones who are earnest about fulfilling their Constitutional obligations.
With respect to Lakin specifically (which is ostensibly what this thread is about), the judiciary to especially include the military judiciary has no authority to inspect the bona fides of the Commander-in-Chief. So long as the office he holds empowers him to issue orders, those orders are presumptively lawful unless they are plainly unlawful, facially.
Such as being by assasination and usurpation? I say, even if they can't remove him, they can declare that he's not President for failure to meet the criteria laid out in the Constitution. Be that criteria failure to be a natural born citizen, or failure to be selected through the electoral college process.
The Constitution gives Congress the power to impeach a President for treason, bribery and other high crimes and misdemeanors. I don't see "not eligible" in that list. With good reason, someone who was not eligible presumably is not President, and thus not only does not need to be impeached, cannot be. You're, in effect, adding meaning that is not there. You're also assuming that inauguration is like a coronation or an investiture. It's not, in fact other than the the President taking the oath of office, it's strictly a traditional and symbolic ceremony, otherwise not provided for by the Constitution. As we've seen before, even the oath taking can and has occurred outside of the inaugeration ceremony. A person becomes President when they meet all the criteria laid out in the Constitution, and the time and date speficied therein arrive. If they don't meet all the criteria they don't become President, de jure.
But what you are really saying is that the courts have no power to enforce the Presidential eligibility criteria of Article II section 1 of the Constitution.