"My question is; If I add some weight to the end of the beam where the double-nuts are, until it sits smack on zero, will the scale measure correctly?"
Yes.
It appears you are trying to zero the old scale with a pan and hanger from a Lee scale. The origianal pan and hanger simply weighed quite a bit more. Make the beam level with the left side screw, hang your pan on it and add enough weight to the nuts to bring it to a rough zero, finininsh with the leveling screw and all will be well.
It's a dead-on accurate scale but you will soon see why men stayed awake at night trying to think of a way to quickly stop the swinging without affecting accuracy. Magnetic dampening was the correct answer!