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"Rationality" is perhaps a more flexible concept than what you're assuming. As long as there's patterns to reality then there will be ways to come up with rules to describe them, and make predictions about future events.
Causality, in particular, is not strictly necessary for rationality. Our understanding of the laws of physics already accounts for situations with uncaused events, in fact. Radioactive decay is an example; an unstable atomic nucleus sitting alone in space with no outside interactions will sometimes spontaneously decay with no preceding "cause". Virtual particles are another - subatomic particles can spontaneously pop into existence and then pop back out again without a specific event causing it.
There are also proposed laws of physics for handling the concept of time travel that can allow for causal loops - things from the future affecting their past selves. We have no reason to believe that this is actually possible, but if one day we discovered that it was possible then it can still be accounted for using rational means.
So I'm not really sure it's possible to have a universe that isn't wholly rational. Even pure chaos is still describable and predictable in its very unpredictability.
With the first example, is that a case of ambiguous causation vs. uncaused? In other words, there may be a cause that isn't yet known? In the latter case, have these been observed, or is this a case of what is supposed to fit/support the model? Not to say it doesn't happen, but that like the former, it's a sufficient explanation given other observations/data in the absence of more information?
That said, I think for some(many?) your description/conception of rationality may be more applicable, but in the terms as I've laid them out, I would call that flexibility a case of rationalizing to accommodate a potentially more unwieldy reality. However, that's to be expected I think, as it seems to be another adaptation in human consciousness to try to navigate being conscious.
It's uncaused as far as we can tell. It's always possible that future evidence may come along to add to this understanding, but without some kind of evidence you can't just make up stuff and call it real.
Flexibility is a necessary part of rationality. To use a real example from history, Newton's laws of motion explained the motions of pretty much everything we could see around us. Objects on Earth, the orbit of the Moon, the orbits of the planets, it all seemed to fit nicely. But then it was noticed that Mercury's orbit didn't quite match the predictions that Newton's laws were making. If there was no flexibility in our understanding of the universe, what should we do? Pretend Mercury wasn't breaking Newton's laws? As it turned out, we needed Einstein's newer more elaborate version of the laws of motion to account for Mercury's motion. Science always needs to be prepared for the possibility that something new will come along that doesn't fit our existing understanding and be ready to expand our understanding to account for it.
So if for example one day we discovered that putting three apples, a digital watch, and an ingot of tin in a row caused a duck to poof into existence seemingly out of nowhere, scientists wouldn't throw up their hands and declare that science had failed and the universe was irrational. They'd start testing whether the species of apple or the time the digital watch was set to made any difference in the breed of duck that manifested.