@Helge: Wow, I think I nerdsniped you into a lengthy investigation!
Thanks for sharing your results here. To be sure I understood you correctly: You think the main component at fault here is the pair of metally-silvery springs going from the center towards the spool flanges and that is what you would like to shim?
I would not be surprised. If you are not investigating closely it might be hard to tell a film flatness issue from a lens sharpness problem. In the camera where your film is, poetically speaking, slack as fuck, can you observe the same kind of film movement as I do in post #13?