If you use a 3D+ crank you don't have to use an adapter in the synapse, and it will still fit your BSA bike.
Unless there is anything else in the bottom bracket on the BSA bike, like Di2 wires, because the 30mm spindle is very close to the ID of the BB shell...
Ignore Amey and get 3d+ - it works in BSA frames too.
As long as you don't have wiring or other nonsense kicking around in there.