The clause you quoted is generally known as the Supremacy Clause. It says that, if a state makes a law or changes its constitution in a way that conflicts with the Constitution, then the Constitution is binding.
"Shall be bound" is the strongest legal language -- where there are conflicts between the actions of the states and the Constitution, there is no choice: the Constitution is the ruling document. Obviously, a state action that totally denies the Constitution is covered by that clause ... it's simply not allowed under the Constitution.
Not to mention that such actions would be "insurrection," in the suppression of which the Federal Government is explicitly empowered (Article I, Section 8) to employ force.
There is also the final clause of Article VI,
... the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution...
Again, "shall be bound" -- and as applied to officers and legislatures of the states, they are not allowed (and the states ratified this) to go against the Constitution, as secession definitely does.
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Nope, not, never... There was no "insurrection" - There was Secession. Did the State legislatures ask for help ? No, they wanted out, and I don't blame them. The whole " republican" form ceased.. when the "agent" invaded the Sovereigns. So much for that "invasion" part too.
Section 4.
The United States shall guarantee to every state in this union a republican form of government, and shall protect each of them against invasion; and on application of the legislature, or of the executive (when the legislature cannot be convened) against domestic violence.
The final clause of Article VI is not in perpetuity. That oath only applies if you are still part of the union. If you separate from the union, then you would no longer be held to that oath.
In marriage, you promise to be loyal to your spouse, and even take an oath “until death do us part”. However, if you get a divorce, you would no longer be held to an oath of loyalty, nor would anyone expect you to. To separate, the parties involved negotiate a settlement, which is what would ideally occur after secession. Forcing a state to remain in the union would be like forcing an abused spouse to stay married. The federal government is in clear violation of its “marriage contract”.
However, if you wanted to get technical, Texas secession from the United States was done by delegation, which is how it will likely be done again. People (on a county-by-county basis) will vote to select delegates for a secession convention. These delegates will decide whether to secede. This takes the state legislature out of the picture, if you were so concerned with Article VI.