Sadly, as long as the North isn't trying to reunite the country, this rump South (look, the South is part of OTL US and splitting up the banking system with 2 currencies is guaranteed to make the South's economy smaller) could go far with a few decades. Mexico and Cuba maybe. USA united could have done a LOT of colonization united (I know Walker failed with Nicaragua, but he didn't have to fail), although if it gave its conquered people the vote, they basically would kill their own government. CSA would be a lot more limited, but the status of the Latin America Republics was not powerhouses. The question is, who is footing the bill? USA in OTL didn't conquer Nicaragua due to impossibility but because they didn't throw everything they had at it, due to not being at risk of bisected if Nicaragua won.
I imagine the US wouldn't sit by idly while the CSA tries to take a few states from Mexico? I guess the timeframe would be very important. Mexico was already finishing up a bloody war of their own but they still had manpower to spare. The CSA...they were getting ready to scrap the barrel. The CSA also has a problem very similar to that of Mexico, little to no power projection. Without the Union's fleet, the CSA would have trouble invading Mexico other than attacking from Texas. They may be able to hold Mexico's northern states, but things get harder as they go south. Unlike in the Mexican American war, Mexico wouldn't mind using guerrilla warfare this time around. Cuba gets harder, there's no land border to cross and then there's the Spanish Fleet. Once again, it's the US that has the navy.
If it happens a few decades after, you'd be looking at a few questions. Did the US still have a Spanish American war? If they did, bye bye Cuba for the CSA or any Central American state. Taking Columbia or Venezuela would be...a bit of a vietnam for the CSA, to use an anachronism. I'm pretty sure the US would also expand the Monroe Doctrine to include attacks from a slaver nation, and they'd get support from the UK as David T mentioned above.
The CSA had big plans, but unless the US avoids Latin America, that's all they can ever really be. At best, taking out the Spanish before the US gets involved and pray that the US doesn't just side with Spain to spite the CSA...I mean in the name of preventing the spread of slavery... That's all they really can do.