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.
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.
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:
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.)