Hello Simon,
I managed one of my rare, brief visits to Lewisham Library today.
I couldn't find anything to link Benjamin with any specific Volunteer unit I'm afraid. The most likely one, provided it was in Lewisham, is Volunteer Training Corps, South London Regiment, 8th (Lewisham) Battalion, "D" (Forest Hill and Lewisham) Company. HQ at 241 Stanstead Road, Lewisham. There were several others though. I looked at the 1917 Kentish Mercury which has a section devoted to volunteer organisations. Unfortunately it is Battalion Orders, so very few individuals are named; normally just the CO, Adj. and duty officers for the week.
I found a picture of the Lewisham Defence League badge and it is a circular enamelled (blue, red and gilt) badge, nothing like a military cap badge.
I also had a look at the directories and there is a Benjamin Cooke at 21 Brookbank Road, Lewisham in 1910 ( the earliest on the shelves). He is still there in 1917. There is no directory for 1918, but by 1919 he has gone.
21 Brookbank Rd is next door to the East London Industrial School for Boys (No.19), which was closed in 1909, although it is still shown as such on later maps. The Industrial School building was destroyed in WW2, while being used as a warehouse. It is at the railway end of Brookbank Rd (furthest from Hilly Fields) on the south side.
Phil