Count me among those who have no idea when Michaelmas is, but does it really matter? The next sentence tells you it is sometime around November. The whole passage is laden with overlapping context clues.
Dickens had a lot of issues with the legal system at the time and it was a protest work.
Toward the end of the story the fighting does stop when lawyer's fees, which they had been charging to the estate, at last empty it. This is announced publicly in court, and the attorneys respond by flinging their piles of paper into the air, one of a few comic scenes in the novel.
FYA, this modus vivendi is still being practiced -- see the litigation around the estate of O.J. Simpson.