I think you are overthinking this one. Just use the linear slide as an air spring, that way you have controllable and constant pressure on the router. A compression spring gets harder to push as it is compressed, so the pressure varies. Using an air spring, the pressure remains constant over the entire travel.
All that's required for an air spring is that the side of the cylinder that you want to use as a spring be plumbed into a small air tank sized so that the volume of the tank is much greater than the volume of the cylinder. This sounds complicated, but it's really simple. Pick up a portable air tank from Harbor Freight, or your favorite auto parts store, and you have a large volume. Supply regulated air to the tank, connect the tank to the cylinder, set the pressure to give you the desired downforce. To retract the cylinder, just supply air at a higher pressure to the opposing side of the cylinder and it will overcome the pressure on the down side, thus retracting the tool. You don't have to relieve the pressure on the down side to do this.
This is harder to describe than it is to implement.