Regardless of when the CSA gains its independence, a lot would depend upon the stance the USA takes toward the confederacy. If the USA is ready to acknowledge the new status quo and develop reasonably good relations with the CSA (which I personally think is the most likely outcome if the CSA acheived independence on its own), successful revolutions would not be particularly likely. The US would not attempt to foment slave revolts or anything else to destabilize its southern neighbor. As long as black slavery survives, the only realistic type of revolutionary activity would be slave revolts. Poor whites were, if anything more racist and anti-black than the slave-owning aristocracy. Unless one wants to imagine some sort of Turtledovian "Super-South", Mexicans and American Indians would be a very tiny minority of the CSA population. Also, it is highly unlikely either group would necessarily find common cause with enslaved blacks. Also, as direct competitors with poor whites as low status freemen, it is unlikely they would join with them.
I think the best likelihood for revolution would be in a more "progressive" confederacy which eliminated slavery and adopted superficially less racist government policies, while still attempting to maintain the aristocratic power base. I'd imagine something like prerevolutionary Russia, but because the CSA would still have a fairly strong tradition of republican and "democratic" attitudes inherited from the USA, even weaker. In this situation, the various poor minority groups as well as the poor whites might join together in some form of Agrarian socialist revolt.