The shutter can't fire until the lever reaches the end of the movement, so the suggestions above still fit with this fact.
Doesn't the AT-1 have the capability to advance the lever a little bit at a time then come back and do it again? In other words, can you cock it with multiple short strokes instead of one long one? If so, and if the shutter fires when it's finally fully cocked, then you've proven it's one of the above suggested problems, and not something physically related to the lever being at its far end position.
Duncan