Alex, per the description below as well as their website YES.
The JP Enterprises Small Frame Variable Mass Bolt Carrier Group Enhanced Bolt is not compatible with folding stock adapters.
The JP Enterprises Small Frame Variable Mass Bolt Carrier Group does not divert gas but can be adjusted to function with a suppressor. The JP Enterprises Small Frame Variable Mass Bolt Carrier Group has adjustable weights to fine tune the cycling of the bolt with a suppressor.
The JP Enterprises Small Frame Variable Mass Bolt Carrier Group Enhanced Bolt is full auto rated. So it will work with the Franklin Armory BFS-III Trigger.
Yes it will work for that application.