Hmm. If you want blacks to become a majority in the CSA, you're going to have to make some rather substantial demographic changes. Whites outnumbered blacks roughly 2 to 1 in the antebellum South. Black and white fertility rates in the USA have tracked reasonably closely throughout the nineteenth and twentieth centuries, and it would take some extreme social changes for them to change enough that blacks can outbreed whites.
I'd suggest three things. Firstly, the independent CSA should be the original seven states, not the eleven it became later. There are a couple of reasons for this. About the only plausible way for the South to win its independence is if it doesn't have to fight for it. Also, the original seven states were those where the black population was largest, and hence they will come closest to forming a South Africa type situation over time.
Secondly, the CSA should not be a target for immigration, at least not white immigration. There are several ways to achieve this. The South was not overly fond of immigration anyways, due to perceived problems with foreigners. An independent CSA should be stubbornly proud of having the "right" view of race - whether slavery or just some version of apartheid/Jim Crow - and mistrustful of foreigners who have different views. Whether through official immigration controls, or just general hatred and harassment of immigrants, the CSA should be the last place on any would-be immigrant's list. It would help if it were an international pariah as apartheid-era South Africa became, but even if not that, it should be hostile to white immigrants.
Thirdly, the CSA should be a target for white emigration. Poor whites should have little opportunity for jobs or social advancement within the CSA, and so leave in search of a better life. The USA is right next door, and won't mind white immigrants. It will, of course, be extremely hostile to black immigrants, so blacks will be stuck in the CSA while whites leave. The ripe time for this is in the late nineteenth and early twentieth centuries, when the USA is going through a massive industrialisation program, and welcoming of any immigrant labour. White Confederates would be ideal... they already speak English, aren't cuturally that different, and will find good jobs for themselves "up North." In OTL this was the era of the Great Migration when a lot of blacks moved north; in an independent CSA, it should be poor whites who lack opportunities at home, and want to leave.
As a consequence of this, the CSA must remain stubbornly poor and agricultural. There may well be some industry, but what industry there is will be in large part staffed either by slaves, or by cheap black labour if and when the slaves are freed. Even then, there shouldn't be too much industry, since that will need a lot of white labour, which is likely to attract immigrants. Fortunately, a lot of the areas which would be likely to industrialise (Virginia, Tennessee, North Carolina) will be excluded if the CSA consists only of the original seven states, so that's another reason for letting those states go.
I'm envisioning a CSA which wins its independence without a shot being fired, and which consists mostly of the cotton states of the Deep South. After going independent, they grow nothing but cotton, cotton and more cotton (except for a bit of cattle in Texas, and later oil, and a small amount of citrus in Florida). The international price of cotton slowly declines, but the South is unable to get around to putting up decent tariffs for a long time. Come the twentieth century, the boll weevil hits, and international cotton prices are at record lows. International competition from cotton grown in India, Egypt etc is increasing, and the South doesn't know much about what they can do with it. No-one who matters in the CSA is going to listen to a black man like George Washington Carver who has ideas about alternative crops, after all.
Poor white farmers have gone bankrupt due to the cotton bust and boll weevils. They first flood the cities in search of jobs, but with Confederate industry moribund and plenty of cheap black labour around, they don't have much luck. The Confederate government remains in the hands of the propertied classes, who prefer cheaper black labour, and while populist movements arise, they don't have much success. At the same time, the USA is turning into an industrial superpower, and crying out for more labour. Poor white Southerners have tried to vote for a government which will give them relief, but since in the CSA it's the people who count the votes that matter, populism fails. Disadvantaged whites vote with their feet instead, going north by the hundreds of thousands. The Great Migration took seven million black Americans north in OTL; a similar white migration could move five or six million whites to the USA and leave the CSA in a rather odd demographic position.
As a bonus, involvement in some sort of major European conflict in the twentieth century would also help. WW1 as we know it is probably out, but there's no reason why a big conflict couldn't start up in Europe, and the CSA could easily get involved. It's not as if they'll be conscripting blacks, so you will need firstly a lot of white war dead, but also some disillusionment amongst the white survivors. They feel betrayed by their government, angry, and that they have no future within the CSA.. and so a lot of young whites leave for a better life within the USA or elsewhere around the world.
Come the 1950s, the CSA remains mostly rural. They've started to experiment with more crops by then, but under the control of planters who remain the effective landlords of whole regions. The middle and lower class whites have mostly fled north, leaving an increasingly outnumbered upper class of whites, and a growing majority of blacks in the South.
How does that sound?