Rotating skyhooks are generally designed to operate from lower altitudes than fixed, to reduce the tether length. This means both that fuel will be being expended to transfer construction materials between the two, and that they will be in different orbits. All orbits around a single body have two intersecting points in the ground track, but normally are different in altitudes which avoids collisions.
Since both of these skyhooks are longer then the difference in altitude this makes collision avoidance a problem. It is certainly possible with careful planning but will increase costs. Most designs for both assume an equatorial orbit, but since that guarantees collision one (presumably the rotating one) would need to be inclined and accept the costs of plane change both in earth relative motion for launched cargo and skyhook to skyhook transfer.
Note also that failure to avoid collision probably triggers kessler cascade so everybody on earth will want to have input possibly leading to bike shedding.
Also relevant is that using a skyhook for bulk mass movement is not free, lifting a mass up the structure drags the rest of the structure down and fuel will need to be expended to bring it back up. This fuel burn can be more efficient than the classical final stage of a chemical rocket, but is still a system cost that means building a rotating skyhook does not automatically make building a fixed skyhook or full space elevator free.
A lot of the very hypothetical assumptions around these structures assume building a rotating sky hook is for allowing access to asteroid or lunar resources, and that these resources would in turn be used to sustain and/or expand the skyhook/space elevator capability.
So the concept of stepping up the lifting capability is often assumed, but generally not directly as 'build rotation skyhook->build stationary skyhook->build space elevator' but instead use the resources made accessible through each step.