I dunno, like, there's nothing saying a freedmen's republic would ever have to be one hundred percent Black, and considering the fact that
large parts of the area were majority enslaved before the OTL Civil War (including nearly the entire state of South Carolina), I think the demographics work out significantly better than you might think. That's not to say that there wouldn't be population movement: I can see a lot of White plantation owners emigrating after the confiscation of their lands and emancipation of their human "property", plus non-slaver White people uncomfortable with the idea of Black majority rule leaving for the whiter Upper South; there would also certainly be large-scale movement of Freedmen from further north seeking a better life somewhere they would be guaranteed to be enfranchised. I just don't think any of that at all necessarily requires "large-scale ethnic cleansing of White people".