It wouldn't be the weight of the spring that has to be negligible, but the change in weight of the spring. Also, the gain in weight the spring would experience wouldn't be equivilent to adding a weight onto the bottom; the added weight would be distributed throughout the spring, and so its impact on the accuracy of the spring would be some fraction of the weight gained.
So if a 1 lb spring gains .1% heavier - even if you consider that weight gain as not being distributed, but just hanging from the bottom of the spring - that right there is only a .001% gain in mass of the system (including 100 lbs of weights). And if you consider that the weight gain of the spring is distributed equally throughout the spring, you're looking at more like .075% effective gain on the spring, and so only .00075% system gain due to the scale.
Even a 10 lbs spring might end up contributing .0075% to the system, 1/16th of the expected change. 1/16th resolution is more than enough for proof of concept.