As others have said. Omega-talk is possible in a purely classical world, and is clearer in a classical world. Omega simply scans my brain and deterministically decides whether to put the money in or not. Then I decide whether I take one or two of the boxes. To say my choice should not have any causal influence on his choice is misleading at least. It may be true (depending on how exactly one defines causality), however it doesn't exclude correlations between the two choices simply because they are both consequences of a common cause (state of my brain and the relevant portion of the world immediately before the scenario begun).

There is no need to include quantumness or even MWI into this scenario, and no certain reason why quantum effects would prevent it from happening. That said, I don't say that something similar is probably going to happen soon.