Thanks Tony,
I have looked through various books on the matter of post war British Military badges, and, in fact, I wore the star as a qualified B3, B2, B1 driver myself in the 90s. I never saw an encircled drivers badge, but I have read about one. My memory fails me as to where I read it, but it was discontinued shortly after introduction. Why, I cannot remember and so I am vexed by this oddity!
Even the various posters that were once rather popular do not give a hint as to what it could be. As you also say, wearing badges is and was strictly controlled, back then even more so, so it has to be an official period qualification badge. Frustrating.