That assumption literally changes the nature of the problem, because the offer to bet, is information that you are using to update your posterior probability.
You can repair that problem by always offering the bet and ignoring one of the bets on tails. But of course that feels like cheating—I think most people would agree that if the odds makers are consistently ignoring bets on one side, then the odds no longer reflect the underlying probability.
Maybe there’s another formulation that gives 1:1 odds, but I can’t think of it.
I don’t see how this is more of a risk for a shutdown-seeking goal, than it is for any other utility function that depends on human behavior.
If anything, the right move here is for humans to commit to immediately complying with plausible threats from the shutdown-seeking AI (by shutting it down). Sure, this destroys the immediate utility of the AI, but on the other hand it drives a very beneficial higher level dynamic, pushing towards better and better alignment over time.