OK, I uninstalled Groupy using RevoUninstaller Pro to clear all remnants, rebooted, and ran the Groupy 2 direct download executable (Groupy2_2.3.1.0-j179-Setup).
I activated it with the Groupy 2 product key for that download, and after it activated successfully, I got the same result as before - a message about requiring me to be a member of Object Desktop Labs, which then opened the Object Desktop Insider information page. When I returned to the Groupy UI, it asked me if I wanted to uninstall it. When I clicked 'No', the UI went away.
The basic Groupy functionality seems to be working, but I have no access to the main UI to configure it. If I open the configuration from 'Settings' in a window menu, it always asks me if I want to uninstall it, and closes if I click 'No'.