Hello,
Once Enterprise Reporter is configured, the security identifiers of the groups are used during role checks, so the behavior of the checks should be okay. I believe your issue is with the name of the groups, where your group rename is not picked up by the server and therefore the group names are incorrect in the consoles.
Without updating your database manually, and changing the cached group name, can you run the Enterprise Reporter DatabaseWizard run an upgrade, and change the group names to the new group names. This will update the cached names, so the consoles will show updated names.