The final requirement submitted for No.2 grenades was 12 January 1916 for 12,000 "Grenades No.2, hand, dummy, with dummy detonators". The final corresponding contract for No.2 was for 12,000 from the Brilliant Arc Lamp and Engineering Co., delivery beginning 11 March 1916 - this was the only contract Brilliant Arc had for the No.2. (All previous contract builds were by the Cotton Powder Co.) So a white painted No.2 with no other markings than BA might be made by Brilliant Arc.
However, it is complicated in that B and A occur on No.2 components in grenades marked Cotton Powder Co. Ltd - sometimes both B and A, sometimes only a B, sometimes only an A, as in the attached.
My take is that they are inspection marks, and given that the No.2 was effectively hand crafted and cost so much, it is not beyond reason to have some parts inspected twice, particularly parts that had to work smoothly together, like the centre tube and needle pellet. The No.2 top body plug shown has both B and A, but as distinct separate stampings, and even on the centre tube the B and A are not in line.
Tom.