I tried moving my movie to a new iMovie project file, tried repairing permissions, tried reinstalling QuickTime, tried restarting, but I can't get this to work.
Sorry, but I'm a little confused at what exactly you're trying to do. You've referred to both a "movie" and to a "imovie project file", in different contexts.
The error occurs when you try to export the iMovie project Timeline to a new QuickTime movie, right? If so, then normally the error message like that would be referring to the folder/disk you are exporting TO. Either the disk is full or you don't have permission to save the exported movie there.
It's possible too, I suppose, that the error message is a bit confused, and the permissions problem is with the iMovie project package (which is why I suggested that.)
Yet the wording "The movie could not be sent to QuickTime, because of an unknown error." seems to suggest the problem is in SENDING the movie to QuickTime, not saving the exported movie to disk. In other words, it may be having trouble GETTING and SENDING the project data to QuickTime.
Maybe it's an authorization problem with the iMovie project music? Does the iMovie project contain iTunes music?
If so, you might try turning off the music track checkboxes, saving the project, then trying to export again. If it exports properly, that's the problem.
If you have QuickTime Pro, you might also try using QuickTime Player to export the "Timeline Movie.mov" that's inside the Cache folder of the project. QT might try to do it differently than iMovie, and not experience a permissions problem, if that's what this is.
If the project permissions look okay, then I'd rule some sort of disk problem — the disk that should receive the export. Check the disk format to be sure it is Mac OS Extended. And, of course, that there's enough free disk space.
Anything else you can say about the history of the project — I notice you describe two different Macs — would be helpful. Has the project been moved from one disk to another? Has it at any time resided on an external drive that was not formatted Mac OS Extended?
Karl