Hi Jeff
I can't see any problem with synthetics, I have used automatic transmission fluid in motorcycle gearboxes, particularly BSA and Triumph in the past with no ill effects, so i can see no problem with using it in the Douglas, which isn't particularly powerful, runs the gears at engine speed unlike most chain-driven gearboxes - so transmitting lower torque, and appears to be fairly robustly constructed.
However, I have no specific knowledge of Douglases at this point, having only just acquired one for the first time.
Cheers
Dave