I've used the cas.fulleditmode cheat to edit an existing family. I've spent hours changing their looks, traits, names, adding family members, etc. and I really like the result. However, my Library still shows the original unmodified family (maybe because I've used fulleditmode to unlock the CAS), so now I can't make a backup, in case something goes wrong or when reinstalling the game

Is there anything I can do to export them? I've tried locating them in the Tray folder and even performing a binary search for their names, but apparently they're just not there. Any help would be MUCH appreciated!

Just a thought, don't know if it is right, but have you tried to give them a different name? Maybe you can't save two families with the same name.
Oh, another thought: when you're in the gallery, did you activate "show custom content" ?

Yes, the CC option is active and I also tried renaming them but they just won't show up in the library. Since families are automatically added when you create them, I assume editing them via CAS cheats won't trigger the library update. Thanks anyway.

When you "edit" a sim that is already in your town all it does is edit them for that town. The sim you originally created is in your library with no changes made. If you want to keep an edited sim you can't change them after they are placed in the town. The only way to change the sim in your library is to go to the library and choose the sim you want to edit. Make any changes you want then save the sim. The changes should be in the library now.

That was it. I feel so stupid for having missed that... (can you believe I had never used that icon before?)

Thanks Warren Zevon & Persian!