What steps would one take to get more actionable information on this topic?
I'd suggest starting by reading up on "brainwashing" and developing a sense of what signs characterize it (and, indeed, if it's even a thing at all).
For each new visitor who comes in and accuses us of messianic groupthink how far should I update in the direction of believing them?
Presumably this depends on how much new evidence they are providing relative to the last visitor accusing us of messianic groupthink, and whether you think you updated properly then. A dozen people repeating the same theory based on the same observations is not (necessarily) significantly more evidence in favor of that theory than five people repeating it; what you should be paying attention to is new evidence.
Note that your suggestions are all within the framework of the "accepted LW wisdom". The best you can hope for is to detect some internal inconsistencies in this framework. One's best chance of "deconversion" is usually to seriously consider the arguments from outside the framework of beliefs, possibly after realizing that the framework in question is not self-consistent or leads to personally unacceptable conclusions (like having to prefer torture to specks). Something like that "worked" for palladias, apparently. Also, I once described an alternative to the LW epistemology (my personal brand of instrumentalism), but it did not go over very well.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.