I remember the official faction rights, may override the overall server rules - if you peek into Official Factions Rights & Responsibilities section and Faction Creation Rules, it hints us the multiple ID (official) faction is allowed.
HOWEVER, I do believe that right is mainly about usage of both Generic ID and Official Faction ID by the very same faction, in my opinion serving as a promotion/restriction item within said faction - because usually Official ID comes with unreachable for generic faction pilots tech, ships, etc.
An official faction can only use their factions ID.
If they want to have a second ID, they must SRP it.
In this situation, the BDM can only use their BDM ID. In order to have any separate second ID, such as the Wild ID for infectees, it needs to be SRPed.