If you take the link in my initial post, there's an extensive section on the town's history. Here is a snippet:

"By 1750 it had become a popular sea spa frequented by Royalty, Poets, Writers and Painters."

Among the writers and artists who visited the area are Henry Fielding and JMW Turner, so it sounds as if it was a popular and well-known place during Jane Austen's time. My favorite quote about Lyme Regis from another writer is by Alfred, Lord Tennyson:

It is said that when Tennyson visited Lyme his friends were anxious to point out to him the reputed landing-place of the Duke of Monmouth, "Tennyson waxed indignant, 'Don't talk to me,' he said, 'of the Duke of Monmouth. Show me the exact spot where Louisa Musgrove fell!'" (Quoted from an Article in Monthly Packet (1893), by John Vaughan.)