Partial answer for now.
tl;dr: It will be an "exercise in futility".
This is a great question! With tens of thousands of very similar satellites in LEO the number of occultations that will occur from a given observatory will be much larger and better characterizable than with the current diverse population, so "if it were useful it would have been tried already" is not a valid dismissal.
However, in LEO they are moving quite fast, thousands of arcseconds per second of time, so any measurement may be limited by shot noise for small apertures, and since the satellites themselves are of the order of a few meters, for large aperture telescopes the occultation is partial and complex; while the satellite may block light for some part of the telescopes mirror, other parts will still be receiving light.
The speed of a satellite in LEO is about 7800 m/s, so if you have an 8 meter telescope the entire event will last about 1 millisecond. There may be some really interesting way to do some kind of very complicated reconstruction for a very bright object when a spacecraft occultation is observed by a large aperture telescope, since what you have is a complex aperture that moves through a variety of states over that one millisecond.
During that time the Earth's atmosphere will be in a fairly well "frozen" state, so one solution for astronomical seeing will probably work for the entire event
But my hunch is that there will be so many small uncertainties and ambiguities that this will end up as an exercise in futility, and events will be so infrequent and non-repeating that there will be no way to check the method.
Here are some plots of apparent angular velocity of satellites in LEO from 300 to 1000 km altitude. The new, large internet-delivering constellations will be closer to 300 than 1000 km, taken from and explained in depth in this answer.
One degree per second is of course 3600 arcseconds per second of time.