Yes and no. Race is a social construct and absent Jim Crow I'm not sure the same definition of being black would be in place. There was a sizable number of persons who identified as mixed-race on census forms until the early 20th century, when such a distinction no longer made a practical legal difference.
View attachment 643429
This is the Homer Plessy from Plessy v Ferguson, a man who was 7/8 white ancestry but was 'black' under southern state laws. If the white majority feels numerically threatened, they might just tweak with their definitions in order to maintain white Supremacy.
Plessy's legal argument, meanwhile, wasn't a demand for racial equality but rather that the state of Louisiana was depriving him of his property (legal status as a white person) without due process of law. Multiracial persons making those sorts of demands seems likely to me.
The Whites of the Confederacy might also do a variety of far more horrific things to preserve their numerical majority. Eugenics was all the rage in the early 20th century.