Yes, I would back the thread being BA, (I think that I have come across a BA thread on a Benson Height Gauge).
If its a British made measuring instrument, of a few years ago, I would expect to find, BSW, BSF or BA threads, rather than UNC or UNF.
My immediate reaction was "Make one". You can have the head whatever size seems/feels/looks reasonable.
Obviously don't have a 1" knurled head on a 6BA, to prevent shearing it off, but 5/16 to 7/16 would seem suitable to me.
Howard