A possible clue I found was in the book "Mapping an Empire: The Geographical Construction of British India, 1765-1843" by Matthew Edney.
It refers to a man named Thomas Jervis who hoped to succeed Everest as the Surveyor-General of India. It seems Jervis was sending letters and petitions that gave the impression that Everest had resigned.
This caused a "a very public squabble" between the pair, Edney writes.
In reaction to some of Jervis's actions, Everest published materials to air his grievances.
These letters - and the bad blood between Jervis and Everest - certainly came to the attention of London's top scientists, including Herschel.
It is not clear whose side Herschel took. And Edney described the incident as "quite minor in the bigger scheme of scientific politics".