I see that a mode of activation is not important. My test shows that Connections->Settings admin menu item is available in case user can ‘connections_change_settings’ capability.
More, plugin has own role editor – Connections->Roles, available to user who can ‘connection_change_roles’.
Look again if some of capabilities started from ‘connections_’ was missed for a role at subsite. I don’t see another reason without deep investigation of CBD plugin code.