They'll become a pariah in the late 1800s. Brazil abolished slavery in 1888, which is going to leave the Confederacy as more of an outlier. My guess is the Confederacy is pressured into abolishing slavery sometime between 1890-1910 for several reasons. First, this was an era that saw increased coordination of the Great Powers, with the Scramble for Africa, the Eight Nation Alliance, etc. This was the era where international law really took off. Second, the United States. I don't expect the Confederacy and US to be long-term enemies. The US won't become some revanchist robot, once it has given up and accepted Confederate independence, that will be it other than perhaps border disputes and some hot words about slaves escaping north (quite possible a treaty has some guarantees about returning them, but the US isn't going to bother acting on them).. There will be some distrust, but also a lot of trade, giving leverage. The Progressive Era is well-suited for a strong movement to pressure the Confederacy to end their barbaric practices, with the ATL equivalent of Teddy to push it along. Third, there will be internal pressure as interest groups besides plantation owners grow stronger. People on here like to talk about how the South was backward, but that is only in comparison to the North, which was insanely developed for the time, with only the British really coming close. In its own right, the Confederacy had the industry of a lesser great power, although wealthier than them prior to the Civil War.
I don't expect an invasion or something, rather increasing quiet threats and leveraging trade with the Confederacy, until eventually a Confederate administration ends official slavery. Then of course comes the slavery-in-all-but-name and Jim Crow. Treatment of Blacks is always going to be something that causes the Confederacy diplomatic problems. It might be a great power, but it won't be a superpower, unlike the US, so easier for other countries to make a big deal about it. Further, the South in OTL US was the backward part which was easy to ignore, while for the Confederacy, that is the whole of it.