Or more simply:
Player 1 makes a move for White.
Player 2 has the option to switch colors.
Play continues with Reversed Armageddon, where White wins ties.
I think this would work so long as Player 1 is capable of making a bad enough first move to cancel out White’s huge advantage from both going first and winning draws. (And ideally there would be more than one move that’s about that bad, so that the opening doesn’t always start the same way.)
There’s a simpler explanation. You say yourself that the answer to this original question is 1/2:
and that the answer to this slightly different question is 1/3:
People disagree about the answer to the Sleeping Beauty problem because those questions are nearly identical and easy to confuse. And because in almost all circumstances the answer to questions that vary this way is the same, so you don’t need to distinguish them.
Bit of a side note, but personally the distinction I like to draw is between the probability that the coin landed heads or tails (which happens on Sunday, exactly once, and has two possible events, giving the answer 1⁄2), and the probability that Beauty observes heads (which happens on Monday or maybe Tuesday, and has three possible events, giving the answer 1⁄3).