dbone,
My procedure would be to take a small torch tip and heat the head of each little screw. Localize the heat until the flathead turns red, remove the torch and let the screw cool.
Before it is dead cold, squirt each screw with WD-40 or an equivalent.
Use a small left hand drill bit, and drill through the center of each screw. If the left hand bit doesn't buzz them out, use a extractor and attempt to twist them out.
This all sounds complicated, and it sort of is...but if you have all these tools, it would most likely give you the best results.
You can also drill a small pilot hole as near to the center of the screw as you can, then follow that with a very large drill. This will 'pop' the countersunk portion off, and allow you to grab the little stub with vise-grips once the latch piece is removed.
Removing broken or frozen screws/bolts requires creativity for each circumstance..good luck.
Stuart