I keep a couple left hand threaded drill bits in my box, just big enough to barely fit into the holes on the screws I use. They're the same as drill bits but, cut counter-clockwise instead of clockwise.
Put the bit in the drill, set it to go counter-clockwise, and carefully drill into the screw. Typically a couple turns into the screw, the bit catches it right and the screw comes right out.
I've done this probably eight times and it has worked perfectly every time.
That said, from that picture, it looks like the dremel and flat tip screw driver will work fine.
Later