Your last sentence, “Congress should be spending its time fighting this president on the political level, through the budget and lawmaking processes, at the same time its members seek to remind the people (and their colleagues) that Congress has the right and the power, should it have the nerve to use it, to settle such disputes with finality.” … is the correct answer!
I do not believe the U.S. Supreme Court is the answer. It raises, again, the powers that belong to the Executive and Legislative branches. This is where it should be fought out. The Congress can proceed with an Impeachment process, or Over-riding the President’s (expected) Veto of legislative laws.
The people have made their — appeal to legislative action — by the recent elections giving the new Congress, both the House and Senate, to the Republican party, to take command ‘of the Laws’, of and to, the Constitution.
Respectfully, John
(Facebook, author of The Tribute)