.mp4 and QuickTime vs Amadeus Pro

Even worse, if I Ctrl-click, bring up the "Get Info" pane and change the "Open With" app to QuickTime or VLC, on double clicking the file, I then get an alert saying that the file cannot be opened as it is from an unidentified developer.

Try the following... Immediately after being told the app/file can't be opened, open the "System Preferences > Security & Privacy > General" window tab and check to see if the "Open anyway" option is available. If so, use it.

Does anyone have a suggestion how to overcome this idiotic situation?

This may depend on how the "unidentified developer" security trap is being triggered. Normally, when an unidentified app is first installed you override the issue as described above. Thereafter, the app should open normally until the next update by the same developer is installed. However, it is unclear if you are resetting the "Open with..." option for each individual file or for all files having the MP4 extension. It is also unclear if Amadeus Pro is resetting the security trap each time it creates a new file of it keeps traping on the files because you have never overridden the trap previously. In either case, you may wish to post back with your results.

What is so bizarre is that the file was not created by "an unknown developer", it was created by one of Apple's own flagship apps, Final Cut Pro X!

My guess would be that for some unexplained reason one of your systems "spontaneously" re-associated the MP4 file extension with Amadeus Pro as the "creator" app. (This has happened to me with regard to VLC but I'm not sure if it was a similar "spontaneous" system glitch or "lazy fingers" accessing an undocumented keyboard shortcut by mistake.) In any case, manually resetting the "Open With" option should prevent any attempt to open Amadeus Pro in future (or until/unless this should ever happen again) while the "Security & Privacy" option workflow cleared the security trap issue. Basically the security trap prevents apps created by unregistered (non-App Store/unidentified) developers from being opened maliciously without your consent. (Assume you're using the non-App Store sourced app here—otherwise I don't understand why the security trap was triggered.)