OP

I found out more about the 'Constitution will hang by a thread' prophecy. It's specifically from Joseph Smith, although Orson Hyde, John Taylor, and Orson Pratt clarified it and provided some important details. First up, Elder Pratt [Pictured above, and yes he really looked like that]:

[God] will speedily fulfill the prophecy in relation to the overthrow of this nation, and their destruction. We shall be obliged to have a government to preserve ourselves in unity and peace; for they, through being wasted away, will not have power to govern; for state will be divided against state, city against city, town against town, and the whole country will be in terror and confusion; mobocracy will prevail and their [sic] will be no security, through this great Republic, for the lives or property of the people. 

Next up, Joseph for the original prophecy:

The time will come when the destiny of the nation will hang upon a single thread. At that critical juncture, this people will step forth and save it from the threatened destruction.

Next, Orson Hyde:

It is said that brother Joseph in his lifetime declared that the Elders of this Church should step forth at a particular time when the Constitution should be in danger, and rescue it, and save it. This may be so; but I do not recollect that he said exactly so. I believe he said something like this - that the time would come when the Constitution and the country would be in danger of an overthrow; and said he, if the Constitution be saved at all, it will be by the Elders of this Church. I believe this is about the language as nearly as I can recollect it. 

The question is whether it will be saved at all, or not. I do not know that it matters to us whether it is or not: the Lord will provide for and take care of his people, if we do every duty, and fear and honour him, and keep his commandments; and he will not leave us without a Constitution.

And finally, President Taylor:

When the people shall have torn to shreds the Constitution of the United States the Elders of Israel will be found holding it up to the nations of the earth and proclaiming liberty and equal rights to all men, and extending the hand of fellowship to the oppressed of all nations. This is part of the program, and as long as we do what is right and fear God, he will help us and stand by us under all circumstances.

Combined with these and some other apostle quotes, the gist of it is that the US will descend into some kind of "Thunderdome" scenario and the Church will be the only safe place in the country. We will govern ourselves and we will do so by the original Constitution, because the rest of the country will have rejected the Constitution and will hold it 'inviolate' or null/invalid. Given this, my original understanding was wrong, and I'm glad for that because this info makes far more sense to me.