I like the rubber band idea, never thought of that
As far as the screw extractors, you won't find one small enough - I know cause I had a similar situation several months ago.
If you can grab the head, which is unlikely, I always use a needle nose vice grips pliers.
Not familiar with Ricks left handed drill bit, but if that doesn't work you're basically left with drilling the screw out. Select a drill bit just slightly smaller than the head of the screw and carefully drill until the head spins loose of the screw body. Carefully remove the air cooler and then use a needle nose vice grips to spin the screw out of the hole. Works like a charm.
Only problem is you'll be short 1 screw when you put it back together, but that should not present too many problems.