It just occurred to me that the rotation of the great circle ellipse is the same as the latitude it touches on the edge of the sphere.
With a non-zero alpha, the vertical offset of the latitude line, as shown earlier, is:
r * Math.sin(latitude) * Math.cos(alpha)
Which means the rotation of the great circle that touches that latitude is
Math.asin(Math.sin(latitude) * Math.cos(alpha))