Your ideas for varations sound plausible, except, I think, for the rocket. Unless the rocket were confined to a closed box, much of the reaction force would be applied to the ground in general rather than the scale in specific, so I wouldnm't expect the force to average out to the weight of the rocket. Another similar case would be the other familiar thought experiment-
birds flying withing a closed cage. The weight of the cage should be the same with the birds either flying or roosting, but only if the sides are closed.
Jack,
I don't know that the more involved analysis would be effective in this case. In principle I know where you are coming from, but in practice I don't think the throws are consistent enough in amplitude or period to adequately fit any regular oscillation to the data. The FFT shows a definite peak at the frequency of the throws, but the peak is broad and lots ot frequencies besides the harmonics have noticable amplitude.
Another option would be to find the mean weight for each period separately, then find the mean and standard error for these numbers. That seems much simpler and about as effective method to look for significant changes.