Rotating a lead screw with a moving magnet

Yes, it is POSSIBLE but it's not very practical.

Take a look at how switched reluctance motors work. Essentially, torque is generated by the motion of the soft magnetic material to align itself with the magnetic field lines. You're using a hard magnetic material (ferromagnetic), but the approach is essentially the same.

IT would work a lot better if you can introduce some level of asymmetry to your screw cross sectional area, but if you don't want to do that, you can get an effective elongated magnetic profile by aligning your stators with the thread pitch of your screw.