I'd like to think that would be one step over the line for the kind of people who would normally have cheered or at least shrugged off violence against blacks and civil-rights activists, and you would see a "backlash against the backlash", even in the south, which might make a smoother course for the movement in the ensuing 1960s.
But the Birmingham Church Bombings three years later didn't exactly slam the brakes on segregationism, so maybe I'm being overly optimistic. Granted, the church bombing, as far as I know, was not specifically trying to kill children, whereas in the OP, the gunman was consciously thinking "Hey, think I'll head over to that school and mow that little girl down." I'm still kinda guessing that most segragationists would say write it off as "Well, we don't approve of killing kids, but if the federal government and the courts hadn't inflamed the situation so badly..."