So yeah;
the eventual boll weevil infestation, if the Confederate States retains the institution of slavery into the 1890s-early 20th century, would be devastating to their economy. Even if slavery by this time has been abolished because surprise, economics doesn't favor slave-holding societies in the modern capitalist system; the South's main industry is about to get skull-fucked by Mother Nature; which is a serious problem.
Moving onwards, there's also the consideration that the South's
claimed reasons for secession wasn't 100% the "retaining of the peculiar institution"; both contemporaries and neo-Confederates insist it was mostly a state's rights issue, rather than a matter of slavery. But the realism of the situation showed that the CSA couldn't survive without a continuity of some power of the national government -- the CSA would have eventually consolidated into something esque to the USA, or disintegrated due to the centrifuge of the states not wanting to take orders from Richmond.
There's also the fact that eventually, blacks who can read and speak, will eventually start spreading leftist ideologies like communism amongst the
literally oppressed masses, and ferment a socialist revolution against the Confederate government; which can only be suppressed violently, further alienating the C.S. from the international community if they start massacring innocents; or slaves... which would be a problem anyway as by 1900, most civilizations find slavery
kind of abhorrent...
But in the immediate, I don't think the US would move the capital away, unless it was a dire circumstance.