Solution

The obstruction of every point on the reduced cross-section of the light beam witnessed by the observer on the disk is simultaneous. However, this obstruction is not simultaneous in the laboratory frame for any two points with different $y$ coordinates. (By Lorentz transformation, if the difference of $y$ coordinates is $\Delta y$, then the time separation of the obstruction events is $v\Delta y/c^2$.) Consequently, points on the same wavefront parallel to the disk are not obstructed at the same time in the laboratory frame, and the paradox is resolved.