There's got to be some website or plugin or HTML trickery that would let me have something like that, right?

Oh, sweet, naive propinquitine! ;) Short answer: not for free. Sortable things require access to a database and LJ ain't gonna let you do that.

The closest free sort-of equivalent would be creating a delicious account and using the tags for various types of sorting. My own delicious account (which is not for my own fic, though) has tags for fandom, author, pairing, genre, length, and a couple misc ones. So then I can find fics by any of those categories.