But control of the franchise is in the hands of state authorities, not the central government. Any constitutional amendment needs eight of the eleven states to support it, and the states control who gets to vote for that amendment. You've cited the cases of South American dictatorships: did any of them have federal constitutions like the Confederacy?
This is a good point: but federalism could be broken under the right circumstances, let's go through it below
The mechanics for this coup seem almost impossible. The moment that the general starts moving regular troops off the frontier towards Richmond, it's going to be clear what's happening. As a result, the president and Congress will flee and call on the state militia to suppress the insurrection.
First, you don't need too many regular troops to mount a coup: the upper-bound for necessary troops is probably 1000. Coups of the central government generally require you to control the nerve center of the state which amounts to maybe 1 or 2 square miles. You could do this without moving too many troops.
Second of all, I think the only real state that actually matters in the immediate aftermath of the coup is Virginia because it takes too long for other state militias to reach the capital before the junta could make some sort of deal with enough state governments to stabilize a government in the aftermath.
Before you call ASBish the state governors would ever accept this, imagine a scenario for instance: that state governments are going bankrupt and congress is unwilling, or unable, to bail them out.
Those state governments are teetering on the brink of unable to pay for very essential government functions (like the troops or police keeping literal angry mobs from lynching them) and there has already being either actual insurrections or the very real threat of insurrections by either poor whites, blacks or both. The coupsters comes in and essentially agrees to bail out the governments and use federal troops to crush any potential insurrections. Oh and of course, I am the best person to defend ourselves from a potential northern invasion you are all afraid of because (*insert US saber rattling crisis here) because i whipped the damnyankees in the last war.
There would be some kind of concession or promise to "have elections" and lip service of "of course state's rights!", but those would be lip service to overcome doubts. Oh and at least some state governors are old war-buddies of the coup leader (let's just say it's James Longstreet leading it). And there would be at least a significant part of congress (though not a majority) which supports him.
Under such circumstances, can you really tell me with 95%+ certainly that most state governments would not at least passively accept the coup? I mean I don't think it's a guarantee or anything but I think it's perfectly plausible that the coup lasts.
What's more likely than this very centralised anti-political move is that you get radical, locally-organised movements campaigning for their states to do more to alleviate the situation and standing candidates for election: a grassroots rather than a top-down phenomenon fits the Confederacy much better.
There's nothing about this which is exclusive with a coup. Our general would not have came out of now where but probably already being on the political scene for years now. There might have already being popular movements to get him in power (somehow) before the coup.
The mode I'm using for this coup is what in Europe is known as Bonapartism, Caesarism and later on: Boulangism. It's premise is a popular military strongman who is the hero of the masses but who undercuts parliamentary (or congressional) politics and democratic norms while claiming the mantle of popular sovereignty.
One can entirely see James Longstreet carrying out a coup and then winning the next election afterwards if he "fixes" the economy as a champion of the white working class and small farmers against a segment of the planter elite.
On the long run the popularity he has with the average poor white in the Confederacy will be the hammer he uses to destroy the power of state governments to stop him from doing whatever he wants to do.
The long-run fate of the Confederate army might be something similar to the military in imperial Japan or Ataturk's Turkey: a state-within-a-state with special privileges but under the nominal rule of the civilian government.
But they also had 80+ years of constitutionalism under the US, plus several centuries of constitutionalism under the British, on which to build. Was a US dictator particularly likely c.1800?
No, I would say that's because there was no great national crisis which called for it: the two points when the US might have had a dictator was the civil war and the great deperssion. During both periods you had presidents who bent the constitutions and had the potential to establish some kind of authoratarian rule.
And the magnitude of the crisis here is greater than any of which the US has ever faced (even more so than 1861 and 1929), an economic collapse in late 1800s CSA would be accompanied by fears of northern invasion and/or slave uprisings. Not to mention of course the "infection" of the minds of poor whites by ideas like Socialism flowing in from Europe and the north....
So you have a weaker constitutional tradition and a stronger crisis putting stress on the system than anything which happened to the USA otl, does the system break for sure? No, but
can the system of limited central government break? Yes definitely.
Note I don't use the term" democratic institution" here: because the Confederacy even before the coup would not have being particularly democratic: even for poor whites: thus undercutting its resilience even further. One can entirely see Longstreet making the CSA more democratic in some ways like expanding the Franchise to consolidate his power.
They'd be a class traitor immediately, by the sounds of things, if they're both deposing the 'planter elites in Congress' and 'fighting the state government for control of the country'. There really aren't any elites for them left to be fighting by that stage. I can't see any Confederate generals- most of whom, let's not forget, chose state over country in 1861- going against their states, their class and their country in the way you suggest.
That's assuming that the prohibition against Congress appropriating money 'for any internal improvement intended to facilitate commerce' lasts until the late 1800s, of course. It would be much easier to get around it by the states funding railways, Congress making donations for 'essential military railroads', or constitutional amendment when the proviso proves to be impractical.
This entire scenario is predicated on the Confederate government being composed of inflexible, dogmatic morons who are seen as out of touch with modern reality.