OK, so I figured out how to work it. You have to do a HARD RESET [*] when you switch locations if you forget to switch profiles before unplugging. This preserves the profiles, so you can go into the Admin Utility and change them back. It also unfortunately resets the password to "public". Still not happy, but at least I can make it work now.
[*] Hold the reset button in for ten flashes. Then wait for a whole minute while it resets itself.