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What Motte and Baileys are rationalists most likely to engage in?

One option might be 'do the rationalist-ish thing when you're forced to because it's decision-relevant; but when you're just analyzing an interesting intellectual puzzle for fun, don't do the rationalist-ish thing'

 

This is the closest to what I was trying to say, but I would scope my criticism even more narrowly. To try and put it bluntly and briefly: Don't choose to suspend disbelief for multiple core hypotheses within your argument, while simultaneously holding that the final conclusion built off of them is objectively likely and has been supported throughout. 

The motte with this argument style, that your conclusion is the best you can do given your limited data, is true and I agree. Because of that this is a genuinely good technique for decision making in a limited space, as you mention. What I see as the bailey though, that your conclusion is actually probable in a real and objective sense, and that you've proven it to be so with supporting logic and data, is what doesn't follow to me. Because you haven't falsified anything in an objective sense, there is no guaranteed probability or likelihood that you are correct, and you are more likely to be incorrect the more times in your argument you've chosen to deliberately suspend disbelief for one of your hypotheses to carry onward. Confidence intervals are a number you're applying to your own feelings, not actual odds of correctness, so can't be objectively used to calculate your chance of being right overall.

Put another way, in science it is totally possible and reasonable for a researcher to have an informed hypothesis that multiple hypothetical mechanisms in the world all exist, and that they combine together to cause some broader behavior that so far has been unexplained. But if this researcher were to jump to asserting that the broader behavior is probably happening because of all these hypothetical mechanisms, without first actively validating all the individual hypotheses with falsifiable experiments, we'd label their proposed broad system of belief as a pseudoscience. The pseudoscience label would still be true even if their final conclusion turned out to be accurate, because the problem here is with the form (assuming multiple mechanisms are real without validating them) rather than the content (the mechanisms themselves). This becomes better or worse the more of these hypothetical but unproven mechanisms need to exist and depend on each other for the researcher's final conclusion to be true.

I hear you on examples, but since I don't like posts that do this I don't have any saved to point at unfortunately. I can go looking for new ones that do this if you think it would still be helpful though.

What Motte and Baileys are rationalists most likely to engage in?

The implied claim that I took from the passage (perhaps incorrectly) is that motte and bailey is a fallacy inherent to post-modernist thought in general, rather than a bad rhetorical technique that some post-modernists commenters engage in on the internet. From that it should be easier, not harder, to cite real-world examples of it since the rhetorical fallacy is actually widespread and representative of post-modern thought. The government example isn't analogous, as it would have at least been a real-world example and the person in that hypothetical wouldn't be trying to argue that the dysfunctional dynamic is inherent to all government. But the quote chose to make up an absurd post-modernist claim about the sun being socially constructed to try and prove a claim that post-modernism is absurd.

I made my aside because I am a relatively everyday person who is a general fan of post-modernism, or at least the concept of social construction as I've described, and I have a strong suspicion that whatever specific real-world examples the author is pattern-matching as denying objective reality probably have a stronger argument for being a socially constructed than they're aware of. Or at least able to hand-wave as absurd as easily as their sun hypothetical.

This is all just an aside of an aside though, and I somewhat regret putting it in the body of my post and distracting from the rest. People generally do make terrible arguments on the internet, so in terms of sheer volume I do agree that bad arguments abound.

What Motte and Baileys are rationalists most likely to engage in?

I've been Rationalist-adjacent for over 10 years now by my ideals, but have never taken part in the community (until this post, hello!) precisely because I find this fallacy throughout a lot of Rationalist discourse and it has put me off.

The motte: "Here is some verifiable data that suggests my hypothesis. It is incomplete, and I may be wrong. I am but a humble thinker, calling out into the darkness, looking for a few pinpricks of truth's light."

The bailey: "The limitations in my data and argument are small enough that I can confidently make a complex conclusion at the end, to some confidence interval. Prove my studies wrong if you disagree. If you respond to my argument with any kind of detectable emotion I will take this as a sign of your own irrationality and personal failings."

In my reading the bailey tends to come out in a few similar Rationalist argument styles. What they all have in common is that some lip service is usually paid to the limitations of the argument, but the poster still goes on as if their overall argument is probable and valid, instead of a fundamentally unsupported post-hoc rationalization built on sand. I tend to see:

  1. The poster makes an arbitrary decision to evaluate the core hypothesis by proxying it onto a set of related, but fundamentally different, metrics from the actual thesis, where the proxy metrics are easily testable and the actual thesis is very broad. The evaluation that follows using the chosen metrics is reasonable, but the initial choice to even use those metrics as a proxy for the thesis question is subjective, unjustified, and the conclusion would have gone another way had different and arguably just as justifiable proxy metrics been chosen instead. The  proxy is never mentioned. Or if it is, it's is hand-waved away as "of course there are other ways to evaluate this question..."  But assuming that your toy metrics equate to a wider narrative is a fundamental error. Analysis is limited to the scope of what it's analyzing to stay accurate. 
  2. The poster shows their work with some math (usually probabilities) to prove a real-world point, but the math is done on a completely hypothetical thought experiment. Can't argue with math! The entire meat of this hinges on the completely unjustified implication that the real world is enough like the thought experiment that the probabilities from one are relevant to both. But the thought experiment came from the poster's mind, and its similarity to reality is backed up by nothing. There is no more inherent reason why probabilities derived from a hypothetical example would apply to reality than random numbers thrown into the comment box would be, but because there's some math work included it's taken as more accurate than the poster saying "I think the world is like X" outright.
  3. Using Bayesian reasoning and confidence intervals to construct a multi-point argument of mostly-unproven assertions that all rely on each other, so that the whole is much weaker than the sum of its parts. The argument is made as if the chance of error at each successive step is additive rather than compounding, and as if the X% confidence interval the author assigns at each unproven assertion is the actual real probability of it being true. But in reality, confidence intervals are a post-hoc label we give to our own subjective feelings when evaluating a statement that we believe but haven't proven. The second you label an unsupported statement with one of these you've acknowledged that you've left what you're sure of as objective reality. Each successive one in an argument makes the problem worse, because the error compounds. It would be more honest and objective for the argument to stop at the very first doubtful point and leave it there with a CI for future discussion. But instead I see a lot of "of course, this can't be really known right now, but I think it's 65% likely given the inconclusive data I have so far, and if we assume that it's true for the sake of argument..." and then it continues further into the weeds for another few thousand words.  

Obviously this comment is critical, but I do mean this with good humor and I hope it is taken as such. The pursuit of truth is an ideal I hold important.

(An aside: the characterization of post-modern argument in the OP is only accurate in the most extreme and easily parodied of post-modernist thinkers. Most post-modernists would argue that social constructs are subjective narratives told on top of an objective world, and that many more things are socially constructed than most people believe. That the hypothetical about the sun is used as an example of bad post-modernist thought, instead of any of the actual arguments post-modernists make in real life, is a bit of a tip-off that it's not engaging with a steel man.)