Since I do not want to be a mere destructive force here I would like to suggest another possible plot , namely Lancelot the Overreacher. Arrhur himself may be too much of a secylar saint to be played as deeply flawed, but Lancelot, while heroic, is an adulterer who betrays his King. With a little tweaking we could easily fit him into the aforementioned ambitious character who defies his proper role causing problems for everybody dramatic stereotype. While the play might be called the Tragedy of Arthur, it might be about Lancelot. Shakespeare might conflate Lancelot and Mordred into one character so the man who seduces Arthur's wife also tries to steal his crown. A tad domestic I know but then Shakespeare did tend towards that direction. Just a thought.
That makes far more sense as something Shakespeare would write. It's not overtly political, and it has a lot of blood and sex.
Depending on the timing of things, I could see Shakespeare putting in sly little references to Essex, as Lancelot-the Overreacher would be a tad similar to that rebellious Earl. Not enough to have the play outright banned, but the usual sort of small allusion to contemporary affairs he usually did, as in his references to the war in Ireland in Henry V.
"Launcelock, thou shaketh thy crown as 'twere gold. Thou art not Caesar nor Sampson of old"