This is just to confirm that I get similar result – when this specific alpha map is used, it is as if the whole map is black (i.e. completely transparent). It looks like only one pixel from the corner of the alpha map is being sampled.
Until someone with more knowledge of Sprite comes up with a solution (I have never used THREE.SpriteMaterial so far), you can simulate sprites with ordinary material. Here is a demo, use the mouse to rotate the scene. The sprite is always facing the camera because of line 66:
I’m not sure whether SpriteMaterial does not work with alpha maps, or it is just me (and you) missing out some important setting. At least I tried for an hour, but I did not manage to make it work.