If you can characterize the oscillation (location of the poles of the mechanical system), then you should determine if the sample rate is high enough.

Is the mechanical system resonance linear (do the pole locations vary with the amount of weight?

When a weight is placed (not dropped) on the scale pan, can you (by eye) predict where you think the final value will be? What characteristics of the observed measurement clues you about the final value? Can you predict the final value within a partial cycle, or do you need several cycles?

Do you need information about the whole waveform? Could you detect the peak value of each cycle and predict the final value based on the exponential decay of the waveform?

It sounds fun,

Bruce