BOAC,
You are right that the V1 in the QRH is based on balanced field, however what that means is that for a certain weight you need a specific amount of runway to satisfy the balanced field requirement.
For discussion sake lets say that a B737 needs a balanced field of 5000 ft at maximum takeoff weight. If you are operating from a 10,000 ft runway, do you adjust the V1 for the additional runway? (Forget about improved climb and derate). I would guess that the answer is no, you use the same V1.
The same goes for obstacles; neither the QRH v-speed chart nor the FMS provide you with the means to adjust speeds for obstacle clearance. You are required to reduce the takeoff weight and then get the associated V1.
So i believe that V1 is based on weight and not runway length.
Mutt.