Alcsentre Calanice
Gone Fishin'
I have to wonder if the shithole that the CSA would soon become, faced with international pressure, boycotts, etc., would result in the CSA falling apart, and the individual pieces outlawing slavery - possibly as a result of rejoining the US or possibly on their own.
So the answer might be 'never', in that the CSA ceases to exist before slavery there does.
I wonder where this international pressure is supposed to come from. Of course there are going to be powerful international abolitionist and socialist opposed to the CSA's treatment of its black population, but there are governments that will be all to happy to treat and trade with the Confederate States, and if it's only to limit the Northern American influence on the Americas. Napoléon III's France comes to my mind directly, with its interests in Mexico and Panama, but of course also Britain itself.
Also, I can understand why you call the CSA a shithole, it was a racist country built on the enslavement of almost half its population; but it's quite dangerous to let your morale judgement impede on your historical assessment of the situation. As of 1861, the South is still very rich, and accounts for the majority of the American exports. Of course, economic decline will soon threaten the South, as agricultural products continue to lose value relatively to manufactured goods; if, however, cotton prices continue to rise, you might also see some kind of Dutch disease hitting the South, which is, arguably, worse than the first scenario.
But I assume that either of this scenarios won't afflict the South immediately. I'm not an economist, and can't develop detailed models of economic development, but I assume that an independent South would have 20-40 more years of prosperity before the disadvantages of a non-diversified kick it. So the South, or, to put if better, the slave-holding ruling class will have some decades to enjoy its independence. Scenarios that predict instant instability and disintegration of the CSA just aren't realistic in that respect.
Now, can the South avoid the eventual decline as the cotton price eventually falls - that is, can the South industrialize? I think this is more problematic than just asking: Can slaves work in a factory? Of course slaves can be made to work in a factory, and the southern gentry could, of course, invest in manufacture. But the lack of an internal market (remember the wealth is concentrated within the hands of a small ruling class, and up to 50% of the population don't even own themselves, let alone large sums of money to purchase consumer goods) might kill every attempt at industrialization - and while exporting manufactured goods might be an alternative, the South might have a hard to compete, lacking any industrial know-how, especially with USA (and probably Imperial Mexico) setting up trade barriers to protect their own industry.