Since this is speculating about nebulous stuff-- like the relation between government policy on social issues (like racism) and how society behaves-- it's much more complicated than "what if Pickett didn't charge?".
But more fun!
If Mencken's take on Southern White American mindsets can be trusted, it seems a big motivation in areas with large numbers of Blacks was to maintain white supremacy. So if the Federal government really was willing to enforce integration of public institutions from 1896 on, you would have seen an early version of "white flight" from public institutions (like schools), then no doubt starving them of public money ("equally"), while a parallel system of private schools would have come about for those who could afford them. Private splendor and public squalor, American-style.
Wow, to have had education go the way that school systems did in most big cities in the 1960s and 70s -- but sixty years earlier, and in huge swathes of the country. The mind reels, and I say this as someone who likes American public schools.
Thinking about it again, though, around 1900 kids, white and black, were more willing to submit to the full authority of teachers (or else!), so maybe "integration" wouldn't have become a synonym for "mayhem". Maybe it could have been a success, the way integration of the armed forces was. That would have been nice.