Very unlikely. Southerners with this dream weren't thinking it through.
Given the transportation technology of the time, conquest of Mexico, Central America and southward could only be accomplished by sea, as indeed the American conquest in the Mexican War was, after an overland attempt had failed.
For over 50 years a British commitment to opposing slavery had been fundamental. The Royal Navy (not to mention the US Navy) might have had something to contribute to the discussion over whether the CSA would be allowed to conquer its way into a slave empire in Latin America.
Your observation about the lack of CSA sea power would certainly not be overlooked by Confederate leaders as this was the reason they would have nearly lost the war. The development of the Confederate navy would have been task number one beginning in 1864. I seriously doubt the Democratic leaders in the north would have gone back to war to save Cuba in 1866.
I believe that if Lincoln had not won re-election in 1864 and The “Peace Democrats” and struck a deal with the south—the second president of the CSA would have been Lee. I believe he would have freed the slaves—in a measured and less traumatic way. Slavery was not a viable system, and would have died out.