Why does it have to be after TSL? That implies some sort of continuity, where events follow other events. As does the fact that Revan and the Exile even exist. You don't want canon/continuity, so don't have any of it. A standalone story with a Jedi Padawan going on missions might as well just be the Jedi Academy game that already exists.
Even the name "Knights of the Old Republic" implies some continuity, since it's definitely set before the original trilogy, during the time of the Old Republic. That might count as continuity too. Or maybe I'm overdoing your remark about not wanting any continuity at all.
"Sir,we are beginning our search for these rumored 'underwater villages'.They will not stay hidden for long". -Battle Droid Commander OOM-9!