I thought it was winter, to avoid disease and smell, but I don't think I know any better than you.

"Persuasion" refers to Sir Eliot and Elizabeth's visits to London being in spring:

"...thirteen springs shewn their blossoms, as she travelled up to London with her father, for a few weeks' annual enjoyment of the great world."

A quick search for "the season" in JA novels turns up only a couple of vaguely relevant quotes:

Northanger Abbey:

"... there was not a genteel face to be seen, which everybody discovers every Sunday throughout the season, they hastened away to the Crescent, to breathe the fresh air of better company. Here Catherine and Isabella, arm in arm ..." (Had I more patience, I'd look up when Catherine was in Bath - but the Bath season and the London season might not be the same.)

Persuasion:
"... the remarkable situation of the town, the principal street almost hurrying into the water, the walk to the Cobb, skirting round the pleasant little bay, which, in the season, is animated with bathing machines and company ..." but this clearly means summer, not the social "the season."