Thanks for the help. Will post results in a bit.
Let me also give credit where it's due: Acake is the creator of the original XCOM mod, my mod is a alternative to his that includes different balancing, a slightly extended armory, and many aesthetic changes. We've been collaborating for some time for both of our mods.
So, basically, I'd be copying the map folders from vanilla into the overlay's maps subfolder, and adjusting the factions so that they point to the overlay's faction files? And this will solve both problems?
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EDIT: Did what you suggested. Got a crash when I attempted to start a new game to switch factions. Crash log included.
https://www.mediafire.com/?i8qhkxcq7t2yo6e