The only option I saw is online competition and when I tried it it told me "There is no online competition to participate in" or something like that.

Oh wait. Look at that. There is a rating battle option. Heh... Sorry I'm dumb.

EDIT: So! I tried a rating battle and lost. Then I tried another one against a guy from Japan and I defeated 2 of his Pokemon with my Porygon2 so and he disconnected on me. I just shrugged it off as I thought it would count towards my wins. It didn't. Back to free battles.