This is the key. To make a rotating skyhook viable, you need to put the rendezvous point for the skyhook/rotovator as far above the dense atmosphere as possible. Because a skyhook/rotovator can be theoretically stationary at the bottom of its cycle, the speed of the rendevous can be negligible.
You should be able to make up the loss of momentum due to air-resistance + the mass of the payload by pumping solar power into the system. This might even be achievable using electrodynamic tethers, although how that could be made to work in a rotating skyhook would be a challenging engineering problem.
These concepts are described in the wiki page on Momentum Exchange Tethers
…the most significant sentence on that page is probably
"Unfortunately an Earth-to-orbit rotovator cannot be built from currently available materials since the thickness and tether mass to handle the loads on the rotovator would be uneconomically large. "
but other, less demanding options are also discussed.