I voted 1895. Though realistically I think it would be somewhere between 1885-1899. The Cotton industry would probably collapse worldwide sometime in the 1870's-1880's like it did in OTL, and the Confederacy would find itself on the cusp of a cataclysmic depression that would threaten to end the Confederacy as a whole. The Confederacy's hands would be tied and if they truly meant to stay independent, they would need to diversify their economy outside of what the institution of slavery could realistically supply.
This would probably take the form of states individually emphasizing and subsidizing different industries not reliant on slavery, Texan oil assuming they don't secede from the economic turmoil, for example. Perhaps some states would even go as far as to institute state level slavery bans in order to promote supplying white farmers with waged jobs.
This would, of course, bring up the issue of where the former slaves go, and I'd be willing to wager that they would be sent either to Cuba, Santo Domingo, Puerto Rico or Yucatan, assuming that the Confederacy has somehow acquired these states by the 1890's, which I personally don't think is that far fetched. Either that or the sharecropping scenario that played out in our timeline plays out again except on steroids.
Another less likely possibility would be the economy being so crippled that the very fabric of the Confederacy is immediately threatened, forcing the Confederate government to convene in Richmond to begrudgingly admit that the institution of slavery had failed, leading to either an amendment to the Confederate Constitution or a completely new constitution, both scenarios would legally outlaw slavery. As you could probably tell though, those scenarios are unlikely to succeed for a variety of reasons.
Last possibility is a war between the North and South occurring in the late 19th to early 20th century (1895-1905, probably). In this scenario the North devastates the South, but due to cultural and political divides that are judged to be impossible to bridge between the two nations, the North allows the Confederacy to persist independently, but with drastic concessions. One such concession would most definitely be the outlawing of slavery.