Problem fixed I think. I copied my original Library to where I want it, my NAS. That Library was mapped to the AAC files on my laptop, so I had to re-map to the AAC copies I had already put on my NAS. As far as I can see, all the multiple genre info is now present and correct. I still think the AAC files only store one genre, but since I'm using the original Library again (which, I gather, is whether the multiple genres are stored) problem solved.
I lost a few working links in the re-mapping process, but I think I've identified the effected path and fixed it. There are a few more dead links, but I think (stress 'think') this is because not all of my music folders were copied to the NAS. These appear to be folders with particular symbols (e.g. "[", "?"), so this looks like a relatively simple issue to fix.
Thanks for everyone's help. It's really helped me understand how MusicBee and digital music files and Libraries work, though I still consider myself a complete novice :-)
edit:
I just wanted to acknowledge that Pingaware's original advice was, as it turned out, essentially the solution to my problem. scampbll, thank you for the advice regarding Library backups - I will definitely be doing that from now on! And Stephen, thank you for the excellent software. While I used it, iTunes was just a necessary evil. But MusicBee helps me to explore and enjoy my music collection with the flexibility that I need. I think it's the best thing since sliced bread.