I can't screw in the screw against the spring, so there is no spring tension in one arm
Correct me if I'm wrong oh mighty V-brake God, but that screw only serves to center the brake, no? Indeed, uninstalled there is no tension on the spring. But once you mount the pin in the canti stud hole and put the spring behind the pin of the V-brake arm, shouldn't there be tension?
If there is still no tension once the brake pads hit the rim, maybe you can swap/add more spacers in between the pad and the arm, forcing a greater angle on the arm and therefore possibly resulting in spring tension? Not ideal, since you want the arms to be parallel to the brake track once they engage, but might be worth a try.