Although it is only a movie, there is a line in "Gettysburg" that completely captures the reason the "constitutional legitimacy" arugment for the South fails: Tom Berenger as Longstreet says "what we should have done is free the slaves FIRST, then secede." But that goes to the nub of it: the only purpose for secession was not "constitutional rights," or even "freedom," but "freedom to enslave," and there is no such freedom implied in the Constitution, in that the 1st, 2nd, and 5th Amendments ALL de facto prohibited slavery.
I admit that the Constitution contained some contradictions but the weight of everything else IN the Constitution is for human liberty, not slavery (qua property).
Further, perceived PRECISELY that the Constitution was no good without the Declaration, and that it was only "law" so long as it was "dedicated to the proposition" that "all men are created equal." Without that proposition, it was no law at all, merely contradictions. The Civil War was about whose interpretation of the Constitution was right---Calhoun's, who said that the property clause superceded the liberty clauses---or Lincoln, who said that all the liberty clauses DEFINED the property clause.
Thus, it was impossible for the South to "free the slaves first," because the war was not about rights, or the Constitution, but ABOUT THE SLAVES.
The fact remains, the South should have had the right to secede and the North should have respected and honoured that right.