What literary movement does Jane Austen belong to?

The Regency movement, because that is the period of time in which her books got published. Her first book to be in print was Sense and Sensibility in 1811, the same year in which King George III is declared insane and the Prince of Wales becomes Prince Regent (hence the name of the period, Regency).

She is definitely not part of the Victorian movement (which started 20 years after her death), nor of the Gothic writing style (which she mocked in her novel "Northanger Abbey", an obvious parody of gothic novels).

I would never, ever dare call her a romantic, since she made it clear in her writings that she did not avail the movement in the least.

Even though her writings have marriages for love instead of convenience (which was but natural in her days), and some of her stories -like Persuasion and Pride & Prejudice- have attractive, swoon-worthy gentlemen as heroes, the Romantic movement focused on feelings over reason, an idea that Jane Austen deemed quite dangerous (if not potentially disastrous, and when taken to an utter extreme, even ridiculous).

It only takes a read of her novel "Sense & Sensibility" to understand this, but her earlier writings make it even more obvious to the reader (she deliberately laughs at the typical 'romantic' heroines in stories like "Love & Friendship", which is my favorite).