Here you are relying on omega using two ordering systems that we already find highly correlated.

What if Omega asked you to choose between a blegg and a rube instead of A and B. Along with that, Omega tells you that it did not necessarily use the same ordering of blegg and rube when posing the question to the copy.

EDIT: More thoughts: If you can't rely on an obvious correlation between the player labels and choices, why not have a strategy to make a consistent mapping from the player labels to the choices.

The key to winning this game is having both parties disagree. If both parties know the goal and have a consistent mapping process, it would be trivial for them to arrive at different choices.

A simple mapping would be alphabetize the player labels and the choice labels. Player(1) => choice(1), Player(2) => choice(2), Player(n) => choice(n).