I think Barry Goldwater would be the perfect president for this. He was pro-civil rights but felt that forcing businesses to not discriminate violated his libertarian ideals.
He was considered so radical by 1960s standards as to be unelectable, though. Which does point to the broader issue that I don't really see the political sense in the federal government overturning Jim Crow laws (no chance in hell the states do that on their own) without also barring private actors from discriminating by race. It very obviously wouldn't be solving the problem, so it doesn't really work as a compromise people come to. As an endpoint, it's a victory for the segregationists. As a halfway point, it doesn't make any progress.