We have had several discussions about this topic on this forum. According to Handbook 1, the only people authorized to view membership records are the Bishop and Clerk(s). Most permissions in MLS are granted through System Options, and are not related to calling. However, on LCR and LDS Tools, permissions are granted to standard callings. Bishopric counselors are simply not granted such permissions, because their calling does not include management of membership records.
Dana Repouille, Plattsmouth, Nebraska