Don't worry about the Basilisk. It can't exist, because Newcomb's Paradox is unphysical.

Specifically, Newcomb's Paradox is incompatible with a feature of quantum mechanics, Bell's Inequality [0][1][2] which makes it impossible for predictors to fully control how a person will actually choose to act. We could even imagine special "Conway's Coins" which can be flipped like normal coins [3] whenever a person wants a little randomness and wants to be confident that they are not being controlled by a Predictor or Basilisk.

[0] https://en.wikipedia.org/wiki/Bell%27s_theorem

[1] https://en.wikipedia.org/wiki/Kochen%E2%80%93Specker_theorem

[2] https://en.wikipedia.org/wiki/Free_will_theorem

[3] https://en.wikipedia.org/wiki/Coin_flipping