Assuming that by the end of reconstruction any whites not ok with the status quo have fled the two states and former slaves have moved in from neighboring southern states
This. For black Republicans to keep control of a state through and after OTL Reconstruction, black Southerners have to realize they can hold at most one or two states, and "fort up" there.
...to take their place then they should have enough people to assemble a decent sized army to defend themselves.
But there is a quality problem. Blacks outnumber whites in two states, but they lack leadership and experience. Nearly all of them were slaves a few years before, held in a state of enforced passive dependency, including near-total ignorance and illiteracy. They aren't ready to fight. White Southerners, OTOH, have a cadre of leaders and are notoriously combative. That's why, despite the numerical advantage of the blacks, the whites won the power struggle.
For the blacks to win in a state, the black "elite" (the smartest, most knowledgeable, psychologically toughest) have to concentrate there. And I think that such blacks felt strong ties to their home districts. They were reluctant to abandon the only homes they had ever known for the unknown conditions in some other state. The black Republicans in the receiving states would have to make provisions for the migrants: housing and employment.
It would have to be a consciously directed process, and AFAIK no one even thought of it.
If it did happen, or something like it, and the white South wrote off a state...
That state would become a magnet for restive blacks. Most of the whites would leave. (There would be a small group of upper-class whites, the Carpetbagger/Scalawag allies of the blacks. This group might even presume to run the state until the blacks push them aside.)
Politically, it would be a Republican one-party state and probably seriously corrupt. It would be viewed much as the South was viewed in the Jim Crow era - a nasty mess, but nothing to be done about it by outsiders.
What else? Very likely there would be a KKK cordon on its borders, to insure any uppity negritude didn't leak out.
Culturally... That state would pre-empt the energy which went into OTL's Harlem Renaissance. It would also divert a lot of the pressure for desegregation elsewhere
OTOH: suppose Lincolnian "Reconstruction". In the period immediately after the War, Lincoln actively works to stand up the Republican Party in the South, using federal patronage and reaching out to former Whigs. At the same time, he insists that Southern states enfranchise
some blacks. Not all them, which meant a social revolution that whites fought tooth and nail; but enough to get a foot in the door of politics.
With no threat of social upheaval, white Southerners accept improved status for blacks. A few blacks are even elected to "safe" offices, such as one of several seats on the county commission or the school board. There's no KKK or anything like it. Later on, maybe county clerk or even state legislator. Once blacks have filled these positions successfully, the black franchise expands, and the whites accept blacks in more "dangerous" offices.
The Republicans push this, of course, as they are the beneficiaries of black voting. But a generation or two down the road, the Democrats begin competing for black votes, and stop opposing them. (In OTL, Memphis Democrat political boss E. J. Crump began collecting black votes circa 1905.) Many institutions and activities remain segregated, but less than OTL, and more genuinely equal.
By the early 1900s, black political activity is fully unconstrained.
One knock-on from this is that there is no Great Migration of blacks escaping Jim Crow. OTL, the GM shifted Southern demographics, ending the black majorities in Mississippi and South Carolina and expanding white majorities in Georgia, Alabama, and Louisiana. ITTL, those black majorities remain. Louisiana may attract black migrants and flip (it was very close). Florida was very thinly populated then, and about half black. As it develops, it may also attract black migrants and flip (it wouldn't take many).
