if they take into account how much the other cares about those results, using some unknown method, they just might be able to systematically perform better

Systematically perform better, but in this specific situation perform worse? That doesn't sound like a winning strategy on a one-shot dilemma. And the parenthetical patch doesn't seem to fix this problem: if the payoffs are randomly assigned in this one-shot case, and you get the short end of the stick - well, you know you aren't going to get another chance, because it's a one-shot case. Good luck trying to counterfactually mug the player here.