BuddyPress Member Restrictions have an advance option if you want to restrict a user for sending new messages based on their Roles and Member Type. For Member Type, you have to activate the BuddyPress Member Type plugin from BuddyDev.
Here is a screenshot showing BuddyPress Message Restrictions advance options.
1. Enable User Privacy:- If you enable user privacy, Users can select among options like friends only, followers, etc. to receive messages.
2. Enable advance Restriction rules? If you enable the advance restriction rules, you can set the limits by role or by member type