It's already arbitrary large. You want that expanded to match arbitrarily large?

When I say "arbitrarily large" I do not mean infinite. You have some fixed computing power, X (which you can interpret as "memory size" or "number of computations you can do before the sun explodes the next day" or whatever). The premise of newcomb's is that Omega has some fixed computing power Q * X, where Q is really really extremely large. You can increase X as much as you like, as long as Omega is still Q times smarter.

Asking "which box should you pick" implies that you can follow a chain of reasoning which outputs an answer about which box to pick.

Which does not even remotely imply being a perfect reasoner. An ordinary human is capable of doing this just fine.

My decision making strategy is "figure out what Omega did and do the opposite". It only fails to produce a useful result if Omega fails to produce a useful result (perhaps by trying to predict me and not halting).

Two points: If Omega's memory is Q times large than yours, you can't fit a simulation of him in your head. So predicting by simulation is not going to work. Second, If Omega has Q times as much computing time as you, you can try to predict him (by any method) for X steps, at which point the sun explodes. Naturally, Omega simulates you for X steps, notices that you didn't give a result before the sun explodes, so leaves both boxes empty and flies away to safety.

That Omega cannot be a perfect predictor because being one no matter what strategy the human uses would imply being able to solve the halting problem.

Only under the artificial irrelevant-to-the-thought-experiment conditions that require him to care whether you'll one-box or two-box after standing in front of the boxes for millions of years thinking about it. Whether or not the sun explodes, or Omega himself imposes a time limit, a realistic Omega only simulates for X steps, then stops. No halting-problem-solving involved.

In other words, if "Omega isn't a perfect predictor" means that he can't simulate a physical system for an infinite number of steps in finite time then I agree but don't give a shit. Such a thing is entirely unneccessary. In the thought experiment, if you are a human, you die of aging after less than 100 years. And any strategy that involves you thinking in front of the boxes until you die of aging (or starvation, for that matter) is clearly flawed anyway.

Furthermore, there's always the transparent boxes situation. Instead of explicitly simulating Omega, I implicitly simulate Omega by looking in the transparent boxes and determining what Omega's choice was.

This example is less stupid since it is not based on trying to circularly predict yourself. But in this case Omega just makes action-conditional predictions and fills the boxes however he likes.