Sometimes I get a similar problem if the head of a screw breaks. If you take some stainless tube large enough to surround the thread (an old golf club shaft for example) and file some teeth into the end you can cut the screw threads right out by using the tube in a drill.
In your case you would need the tube to be large enough to get the screw head through the hole. There are plug cutters which would also do the job but generally leave a larger hole. A multi tool like fein can cut through the tip of the screw but it takes it's toll on the blade and leaves you with the problem TW mentioned that it will rust.
Sometimes I get a similar problem if the head of a screw breaks. If you take some stainless tube large enough to surround the thread (an old golf club shaft for example) and file some teeth into the end you can cut the screw threads right out by using the tube in a drill.
In your case you would need the tube to be large enough to get the screw head through the hole. There are plug cutters which would also do the job but generally leave a larger hole. A multi tool like fein can cut through the tip of the screw but it takes it's toll on the blade and leaves you with the problem TW mentioned that it will rust.