I don't think it's using a non-causal decision theory that makes the difference. I could play myself in front of a mirror — so that my so-called opponent's move is directly caused by mine, and I can think things through in purely causal terms — and my point about the payoff table having only two entries would still apply.

What makes the difference is whether non-game-theoretic considerations circumscribe the feasible set of possible outcomes before the players try to optimize within the feasible set. If I know nothing about my opponent, my feasible set has four outcomes. If my opponent is my mirror image (or a fellow TDT user, or Omega), I know my feasible set has two outcomes, because (C, D) & (D, C) are blocked a priori by the setup of the situation. If two human game theorists face off, they also end up ruling out (C, D) & (D, C), but only in the process of whittling the original feasible set of four possibilities down to the Nash equilibrium.