How do I remove stripped screws?

In my experience 'defacing' the Thomas engines, those face screw heads are easily stripped because most modifiers do not have a short enough, right size head, screwdriver to get into the narrow, short, engine shell to remove the screw straight on. So most use a normal sized (too long) of screwdriver which doesn't allow the Phillips head wings to fit correctly because the driver is on an angle. Works most of the time but if the screw is too tight the head will strip out as shown.

If you have a Rotary Tool with a thin cut-off wheel/disc you could try slicing a groove in the head and then use a slotted screwdriver. I would also suggest trying to warm up the plastic where the screw is, with a Hair Dryer but do not heat it so much that the plastic gets soft or damage any stickers/decals. You may want to at least try the warm up method first and use the screwgrab/rubberband methods you tried before but be careful with the heat.

I do have a very thin screwdriver that fits within the boiler/tank nicely, but the screws on HiT Toys products are just so tight that it takes an astronomical amount of effort and grip to actually move the screw, and I failed a couple too many times and ended up just wrecking the whole screwhead. I've had similar problems with the screws on other HiT Toys rolling stock, but it's particularly difficult to get a grip because of the delicate nature of the engine and the tight space..

What rotary tool would you recommend? I'm not knowledgeable about hardware (in regards to computers and crafts alike) so I don't know where to start and don't want to end up ordering a tool that will be too big or too destructive for such a small job.