Ah, yeah. Silly Microsoft and their privacy invasion garbage. :/
Big Box unfortunately won't truly prevent access to the rest of the computer. It might be able to trick a young kid, though. You can protect it with a pin and disallow exiting. But of course there are keyboard shortcuts that can still get around it; I don't do anything to force keep you in Big Box.
The best solution is to put everything in Dropbox, including the entire app, your ROMs, and your emulators. It's easiest if you just put everything inside your LaunchBox folder (Emulators in an emulators subfolder, games in a Games subfolder, etc.).
I personally know Dropbox works completely seamlessly that way. Google Drive and OneDrive should as well, in theory. However, the last time I used Google Drive (more than a year ago and I never tried to put LaunchBox in it), it was an atrocious experience, because it crashed constantly. So just FYI on that. Let me know if you end up using OneDrive or Google Drive and how well it's working for those.