BuddyPress.org

Without modification this is what you are limited to with regards to roles and capabilities:

https://codex.wordpress.org/Roles_and_Capabilities

However there are a number of permissions and membership level plugins out there which may offer variations.

https://wordpress.org/extend/plugins/search.php?q=Permissions&sort=
https://buddypress.org/extend/plugins/?search=User

I think perhaps though your sticking point may be admin check without admin rights though.