That diameter seems like an M1.6 screw. I would not want to try to force an M2 into an M1.6 threaded hole, because, besides the shaft diameter difference, the threads have different pitches (0.35 vs 0.40 mm).
I can measure the depth of the hole without much trouble. I have M1.6 x 3mm screw, so I could check fit of the threads with that.
But, another possibility for that diameter is a 0-80 Imperial screw, which has a slightly finer thread pitch (~0.32 mm). I have more lengths of those, so I'll see what fits best.