I've come up with a radical-but-possible explanation for the Special Duties cap-badges (and possibly also for the arm badges).
This could be for at least some of the circumstances (but may have had a wider significance).
Put simply, I think that the RAF Special Duties cap-badge was intended (at least in part) to be worn by Officers
commissioned into what appears to have been an "Administrative and Special Duties" branch of the RAF Volunteer Reserve. In other words, potentially these individuals were not all civilians: at least some were on full wartime emergency commissions.
My research in the Gazette seems to indicate the earliest instance of Special Duties Officers in the RAFVR as being February 1939.
However: there also other officers commissioned into the Admin&SD branch of the RAFVR that are recorded as being awarded "honorary commissions".
I'll attach cuttings from the Gazette, but could I also make a broader proposition (which could either be accepted at face-value, or pursued by looking in detail at contemporary editions of AP1358) that if there was actually a dress regulation that Officers in the Special Duties Branch of the RAFVR were to wear the chrome/silver 'Special Duties' cap-badge, it is entirely-possible that this direction could either have been unofficially ignored (and Special Duties Officers would've then just frequently worn standard RAF Officers' cap-badges, and retained their Special Duties ones as keepsakes); or, any such regulation may then have been officially rescinded (with then the reverse case holding true, and the wearing of Special Duties cap-badges would've become extremely rare- or (conceivably?) been continued only by those listed as having "honorary commissions"?
(Surely there
cannot have been two parallel Royal Air Force associated branches in WW2 named "Special Duties"?)