Changing the Template from within WHMCS Setup > General Settings > General > Template doesn't have any affect on the plugin.
For example. If I change it from Portal (which it should be) to Default I don't see the plugin break or any styling changes. I do see the template changing at my WHMCS installation url.
I've tested this in the base 2012 theme and the problem still exists.
Eg: digitalessence.net / billing shows the default WHMCS template.
and I would now expect the WHMCS/WP integration to break which it doesn't:
It used to to this. I've tried disabling the plugin/activitaing it, changing my WP themes etc.