Technically yes, but there is an issue you will encounter.
That is, when you accelerate your space ship against the counterweight wire, that wire won't have as big inertial mass as the entire space elevator, so it will start to collapse towards the space station in a very complicated manner, considering that any wire that long is flexible and stretchable instead of being rigid and reliable position wise. Therefore, a single launch to a solar orbit via space elevator can render it useless, or at least would require an indeterminate amount of time to stabilize the counterweight for the second launch.
In order to successfully launch something using space elevator as accelerator, your best action would be stop accelerating against it as soon as the bulk of the elevator's mass is below the accelerating device, meaning that even if your counterweight is a wire stretching 36000 km into space, you should NOT use it to apply force, or else the counterweight would be accelerated towards Earth many times stronger than below the station. Also, after you launch, you'll have to accelerate the space elevator itself in order to maintain its orbit, because its structure and orbital parameters are all balanced around having its mass center in relatively the same location, but here you have a free momentum towards Earth for the space elevator that you have to compensate somehow.
This means that if your space elevator has a station at its middle, you cannot effectively use its counterweight to accelerate space ships. Better make your elevator's station, where the bulk of its mass is located, positioned at its very end, a tad above geostationary orbit, yet have its orbital speed in sync with the planet, this way the station would act as both the hub for whatever orbital activities and the counterweight for the wire down to the surface. Such construction might have an opening in the center to allow accelerated cabins to slide off the wire into the open space, although whatever mechanism used to keep the wire attached would need to account for such use, implementing devices to allow detaching from the wire prior to reaching the station. But, speaking of theories, what you ask for is possible with a certain aftermath you'll have to deal with.