This picture from Tanks and Tank Folk by Eric Kennington shows the NCO's arm badge worn by a WO with a backing of similar shape to that on your BD.
It was cavalry and yeomanry practice that these arm badges were only worn on the right arm which raises the question as to what was worn on the patch on the other arm which is in any case of a different shape.
The Westminster Dragoons served with 79th Armoured Division as a flail tank regiment from October 1943 until the end of the war.
Jon
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