In Harry Turtledove's Timeline-191 series, he analogizes the CSA and Nazi Germany. In this AHC, I challenge you to make a CSA that resembles South Africa- a country with a well-constructed constitution but persistent and pervasive inequality- blacks are enfranchised, but most are extremely poor and live in slums or rural backwaters- disease is rampant and whites still hold a lot of de-facto power, though every president since black suffrage has been black, as they are the majority of the country's population by far. Bonus points for having it be a US ally both before and after black enfranchisement, and making it an important regional power.
It would have to be majority black, and even the "Gulf Squadron" (the seven Deep South states that declared secession in December 1860 and January 1861) were not, collectively, majority black (52.7% white). If all the slaves and free colored in the Upper South, Border States, and North were moved to the CSA, it would be about 60% black.
Suppose the Deep South declared secession, and got away with it due to the connivance of President Lane. (Lane was Breckinridge's running mate, who would have been elected VP by the Senate if there was no Electoral College majority, and would have become President if the House failed to name a President. In fact, some Republicans in 1860 campaigned on the slogan "Lincoln or Lane", arguing that if Lincoln did not win outright, Lane would become President.)
Lane blocks all action to stop secession; then suddenly is removed from office in a quasi-Constitutional coup - with the Deep South missing, Republicans have a majority of delegations and elect Lincoln. The CSA has had several months to organize, all Federal forts and dockyards were handed over - it's too late to put down secession. But further secession can be blocked, and Congress enacts several strong anti-slavery laws, including a gradual emancipation amendment (all children born after 1 January 1865 are free).
Slaveholders in the Upper South and Border see the handwriting on the wall, and either sell their slaves south, or move south with them. The free blacks in the USA are subject to much discrimination and persecution. The CSA abolishes de jure slavery around 1885. After 1885, most USA blacks leave for the CSA.
From 1890-1930, a lot of poor whites moved out of the CSA to the USA, leaving a white minority almost exclusively composed of slaveholders and their households, and numbering only about 25% of the population, but having a monopoly of power. This system breaks down in the 1980s; the whites agree to a power-sharing system with the blacks. A few years later the whites are forced aside; but the new black rulers are thoroughly corrupt.
As to the CSA being a "regional power", it would be too large not to be - though greatly inferior to the USA.