I think you're overestimating how much fuel could be saved by this. You need 7.8 km/s of delta-v expenditure in any event because that's orbital velocity. Generally a rocket uses up 10.5 km/s of delta v to reach orbit with the difference being atmospheric and gravity drag[1]. So you'e got almost 3 km/s of delta v you could theoretically save here. You'd probably lose most of the atmospheric component but there's still the thrust you need to counteract gravity for however long it takes you to reach orbital velocity. The lack of air means you can safely accelerate a little faster but you're still limited by the weight of your engines and the heavier structure that faster acceleration requires. I'd put the reduction in needed delta-v at closer to 5% with a fuel reduction around 12%.
Ride the elevator out to geosync, and you'd be in orbit for free. Ride it past there and let go, and you'd end up flying away from the earth. Time it right and you could end up traveling to the moon or to mars for little to no fuel costs, and just have to brake at your destination.
Edit: My mistake, I didn't realize that this wasn't the normal concept of tether out past geosync.
[1]https://en.wikipedia.org/wiki/Gravity_drag