1810s, so well before the whole "inventory of homosexuality" era, which Ulrichs seems a part and parcel of.

Thanks for linking that Upchurch post, that's fascinating--I've certainly seen evidence (a la Bentham) and arguments (a la Randolph Trumbach) that homosexuality was considered by at least some to be innate/natural, and the idea of a gender "inversion" was certainly there (back to the idea of the molly), but it doesn't seem to have been expressed in the term "invert" itself, as far as I can discern.