How do I remove stripped screws?

I've tried the rubber band trick, and I even bought screwgrab, but neither worked, so I'm wondering if those screw-extraction drillbits come in sizes small enough for these trains.
Specifically, I've stripped the screw holding HiT Toys Bill's face onto his smokebox. I'd really like to avoid damaging the plastic at all, and don't want to waste more of my money like I did on the bottle of Screwgrab.

If those drill bits DO come small enough, I don't think I'd necessarily need to replace it anyway, since the plastic parts that actually hold the face in are a pretty tight fit, and the custom faces I want to get will probably have to be applied with blacktack instead of being slotted in anyway.

I've never used a drill before, so what kind would you guys recommend I get for Christmas?

Any better ways I could remove the screw?