It turns out this equality is correct: including the rotation of frames fits in with the angular separation of the composite velocities, and the rotation breaks the symmetry in the overall Lorentz transformation making it nonsymmetric.
It turns out this equality is correct: the rotation of frames matches the angular separation of the composite velocities, and explains how one composite velocity applies to one frame, while the other applies to the rotated frame. The rotation also breaks the symmetry in the overall Lorentz transformation making it nonsymmetric.