Will be interested in seeing your idea and numbers when it's ready for discussion.

IMO, there are two main solutions to the "reboost" problem of a skyhook system:

First, shoot for smaller payloads. If you have an atmospheric skyhook, there is no particular reason you have to shoot for a fifty ton payload.

Second, a conductive tether can be used to boost an orbital object if one can apply electrical power to the tether. This can be done either with a huge solar array, a nuclear reactor, or a power source (solar power station) in a higher orbit that beams power to the skyhook.

Whether this "tether boosting" can effectively be done for something as massive as a skyhook assembly, I don't know. I haven't done any numbers on it at all.