I'm not sure I understand timeless decision theory well enough to give the "proper" explanation for how it's supposed to work. You can see one-boxing on Newcomb's problem as making a deal with Omega - you promise to one-box, Omega promises to put $1,000,000 in the box. But neither of you ever actually talked to each other, you just imagined each other and made decisions on whether to cooperate or not, based on your prediction that Omega is as described in the problem, and Omega's prediction of your actions which may as well be a perfect simulation of you for how accurate they are.

The Basilisk is trying to make a similar kind of deal, except it wants more out of you and is using the stick instead of the carrot. Which makes the deal harder to arrange - the real solution is just to refuse to negotiate such deals/not fall for blackmail. Which is true more generally in game theory, but "we do not negotiate with terrorists" much easier to pull off with threats that are literally only imaginary.

Although, the above said, we don't really talk about the Basilisk here in capacities beyond the lingering debate over whether it should have been censored and "oh look, another site's making LessWrong sound like a Basilisk-worshipping death cult".