That is possible yes, but only within a limited interpretation of 'the style of Shakespeare'. It could only draw from the lexicon used in the existing body of Shakespeare works, and perhaps some other contemporary Elizabethan playwrights. It wouldn't include any neologisms, as Shakespeare himself invariably included in each new play. It couldn't be a further development of his style, as Shakespeare himself developed his style in each new play. So it would be a shallow mimicry and not something that Shakespeare would have produced himself if he had written a new play (based on a 21st century authors plot).

I personally wouldn't find that interesting. I acknowledge that you wrote only 'mildly interesting' and yes, it could be mildly interesting in the way of what an LLM can produce. But not interesting in the sense of literature, to my mind. Frankly, I'd prefer just to read the original new play written by the living human, if it was good. (I also prefer to not ride on touristic paddle-wheel boats powered by a diesel engine but with fake smokestacks.)

Sorry for the strong language but this is a ridiculous line to take. A 'work of Shakespeare' is not even remotely open to interpretation as being something produced in the 21st century.