The real question here is simply "If you experienced something fantastically unlikely, would you take this to be evidence that MWI is true and Copenhagen is false?"

And if so, does experiencing something humdrum constitute evidence that MWI is false? Surely not.

It seems to me that either (a) experiencing anything at all, regardless of how likely or unlikely it is, gives an equal amount of "anthropic evidence" in favour of MWI, whatever that means; or else (b) There is no sense whatsoever in which observations as opposed to a priori reasoning can favour MWI over Copenhagen (pretending the latter is coherent).

I don't think (a) makes sense, because surely any theory whatsoever comes equipped with an anthropic "dimmer switch" that can be "brightened" or "dimmed" arbitrarily. For instance, to 'brighten' Copenhagen we could just stipulate that there are N parallel non-interacting Copenhagen universes rather than 1. We could even let N be infinite.

So that just leaves (b).