How much cable can I pull with a brake lever in another custom mechanism, and how can I achieve it?
I believe a typical V-brake lever (so called long pull) has enough action to pull 2 cm of a cable.  

V-brakes are known for they greed for cable length - e.g. if replacing cantilevers with V-brakes one needs to replace the levers as well.



EDIT: I may be wrong - see this thread: What is the actual cable pull of brake levers?
There it says that long pull has the action of 14 mm which is not enough for you. A solution for you would be (depending on how technical/machinist savvy you are) to use the brake lever to engage another lever that increases the advantage of the cable pull.

Since one picture saves 1000 words...

This contraption would increase the pull of the secondary cable by the ratio of the arms of the extra lever at the cost of increased force required to engage the brake lever. Of course the extra lever can be in any shape and the distance between the rotation point and the cable attachment point determine the arm lengths.

I hope you get the idea.