The only AA light I'm positive is short enough is zebralight H50 / H50b, or possibly H501.... But that's out of your price range. It may also be too thick, but I don't think so. (Depends on the box). You could also get just a tiny bigger box, that would make finding and fitting an AA powered light much easier.

If it has to be such a small box, I personally would have chosen another battery format than AA. Either an AAA-size or a photon light. Or you could pack an AA size light separately, or tape it to the box. If so, get something that you can "lockout" (light won't work if tailcap is loosened a few turns) so it won't click on accidentally.