Does the dimming of Betelgeuse present any observational opportunities? Recent photometry suggests that Betelgeuse has been dimming over the past three or so months, reaching the faintest V-band magnitude seen in modern observations. The changes are apparently even visible to the naked eye, and appear to correspond to a temperature drop of $\sim$150 K relative to maximum brightness. The second Astronomer's Telegram on the phenomenon ends with


  This continues to be an opportune time to carry out complementary measures of Betelgeuse while it is in its current low state and is unusually cool and faint.


This got me thinking. Obviously, continuing photometry will tell us more about the dimming itself. However, regardless of the mechanism behind the phenomenon, is there anything we can glean from the dimming about the star itself - e.g. composition, rotation, limb darkening, etc. - and its circumstellar environment (e.g. a binary companion, which I assume is unlikely, or circumstellar dust)?

Essentially, are there any observations of Betelgeuse that would be made easier by the recent changes? Or am I being overly naïve, and they're really too minor to be of any use?
In terms of "easier" - not really. There isn't anything made particularly easier or more optimal to measure. Theoretically, as you mentioned, "space stuff" like tiny planetoids or dust that are very close to the star itself might become more visible, but that's about it.

However, the reason we would want to get measurements now is to compare them to data collected when the star was at normal brightness and see differences, perhaps in spectroscopic lines, for instance. As well as that, if the star also returns to normal brightness, then we can observe changes that occurred side-by-side with the dimming or maybe even as a result of it.

Side note: Regarding the composition of the star, which you provided as an example - the reason we are unsure about when Betelgeuse will explode is because we can't actually know its true composition, since we can't see directly inside the star. If we could, we would be able to predict when it would finish fusing remaining heavier elements and then extrapolate when it might go supernova. All we can really see is what is coming off of the surface of the star, infer its age from other data, and then make an educated guess about when it will explode.