The CSA's ability to invade Cuba simply does not exist. An independent CSA, already a dubious proposition, has limited industrial resources and a set of feuding and powerful state governments. Hard to put together the resources for a fleet out of that. I mean, look at the battles: the battle between the USS Monitor and the Virginia is a fight between a sizable chunk of the South's naval resources, and a special USN project specifically to defeat that one ship - while still maintaing a blockade. Similar with the fact that the USN could have several ships like the USS Kearsarge combing the seas for the Alabama.
Secondly, populations size. When one third of your population are chattel slaves who know they missed being free by which ever razor thin margin the PoD is, you keep your bayonets at home. Doubly so when you consider that aside from white supremacy and chattel slavery, keeping the lower orders in their place is a big part of the CSA's secession (Seriously, as should be required reading on these boards, read the bl**dy Cornerstone Speech). You've got a fair number of people through the 1870 and 1880s, in this alternate South, getting a lot poorer than their counterparts in the North, because all myth-making aside, the CSA would have been a rigidly hierarchal society with most means of both agricultural and urban production firmly in the hands of a planter elite.
The TL

R here is it takes a lot for a hierarchical slave society (which the ante-bellum South most certainly was) to sustain a large military force in the field. Look at the South's travails during the war. It's even harder for them to maintain a naval fleet. You need both to conquer Cuba, especially when about the one thing the Spanish and the Cuban Nationalists agree on is that your slave-driving assholes.