So no Beethoven (come on, op.22, for sure noone is playing it....play it, play it....) or Mozart?
Now that I think of it, a Haydn sonata may be not as good as an ass-kicking Schubert! I like the idea of the Impromptues (although I would possibly take op.142) OR the Wanderer Fantasy. What do you think of this? I mean, it is a bit longer than 20 minutes and a hell of a piece, until december you will probably fail to learn it, but then I have no idea what your technique is, and, compared with other pieces, there are not so many interpretation issues. Technique is very hard though...(all mov. form a huge compilation of any technique you could think of until Liszt...arpeggios!!!, double octaves, big jumps, chord tremoli (4th mov.!!!), fast runs....)
You could do a sonata as well...but not sure which one...
If you dislike Mozart, then do NOT play him in any case! I mean, K.576 is wonderful, but without a teacher and a deep passion for Mozart, this may be difficult.
Personally I love the Beet. concerto, and since the SS is overplayed as well (perhaps not as much), I say do the Beethoven.
Bach. Yeah, I thought of the Chromatic Fantasy too, but then again, I thought you would probably try something more obscure. Okay, Italian concerto isn't real Baroque style counterpoint, much more "classical" in some sense, so if you like it...
I think you do good in participating without a teacher... quite brave, keep it up!
(And perhaps give some more detailed information on your preferences and last pieces...)