Or what is perhaps more possible: an earlier and somewhat different civil rights campaign, focusing on voting rights and lynching, but not challenging segregation (with all its emotional baggage). Let's say that this starts with the election of Dewey in 1948 over President James Byrnes. While many black leaders are dissatisfied with the limited scope of Dewey's civil rights legislation, it does have the effect of undermining the political hold of the white supremacists. By 1960, under Dewey and his successor, Earl Warren, the white supremacy regime has collapsed politically, and desegregation of government services is underway.
Black votes are divided about 50/50. In 1960, Warren is narrowly defeated by Humphrey, running with Fulbright against the Middle East "police action" and the recession of 1959-60. Humphrey pushes through new, more drastic civil rights laws, and is assassinated by a disgruntled white supremacist. Fulbright refuses to enforce the new laws, outraging liberal Democrats.
In 1964, Pennsylvania Governor Scranton is elected President over John Kennedy, after Kennedy's lurid personal life is exposed. Kennedy's last minute attempt to salvage his campaign by running far left backfires. He is the first Democrat to lose every Southern state. Scranton quietly resumes limited enforcement of the Humphrey Acts. By his second term (1968-1972), resistance dwindles away, as Southerners no longer want the backwards image.
From then on, black votes remain fairly evenly divided. By 1996, race is not a political issue at all. In that year, Illinois Governor Jesse White (D) was elected President.