In ASA SP, just strike three, batter's out.
I only call SP, but I'll take a stab at FP and Modified, so tell me if I'm wrong. Since it was a dropped third strike, the contact after the drop might be interference on the batter. I'll leave that to the FP and Modified gurus and fully accept if I'm wrong.
I have seen inexperienced players swing and catch the ball on the backswing before, but only once or twice.