Since you have the lock out, I suggest you take it apart, buy a lefthand drill of appropriate size and drill it out, after soaking with penetrating oil. If you are lucky, the lefthand drill will unscrew it. You mentioned having an end mill, so I'm guessing you have a milling machine that will turn in reverse.
To further refine the two suggestions made above, you could also buy a left hand center drill, which would make this task easier, and reduce the possibility of the drill bit walking off the center of a 2 m/m screw. W.W. Grainger and MSC sell left hand center drills, and the #1 size would have drill tips that are 1.1906 m/m dia.
I also agree with the idea to soak it with a good penetrating oil, and give that several days to get into those tiny threads. I'd dip into my emergency stock of the now discontinued Tasgon for a job like this, but I have found that pure oil of wintergreen is very good on small fasteners as a penetrating oil. About a week ago, I was able to remove the remnants of four 3 m/m screws that sheared off flush with the surface. But they weren't rusted, and I was able to catch the edge with a dental pick, and turn them out far enough to grab a thread with small needle nose pliers to remove them. You may want to try that first, using a very small pointed punch near the edge to get things started counterclockwise. I find I am able to remove a good percentage of broken screws without resorting to drilling.
I find it very surprising that your broken screw seems to be a standard metric thread, and not some obscure bastard size. Good luck.