Ive been something similar on the Frontpage Forum. What I have done is use Autocreate Groups using a Cohort to create a single Group. And then I have restricted access to the Forum to the Group.
There is a plugin called Cohort sync to Group which might do this automatically. But there should be a Cohort option as part of a restriction - not sure why there isnt? I wonder if there is a Moodle Tracker on this issue.