Brian, I wouldn't want to question what your venerable source remembers. All I can say is that a bowling green is not shown on the 1936 OS map (at a scale of 60" to the mile, and showing each individual tree), and it is not mentioned in the 1930s borough guides.

Is it possible that your source remembers informal games of bowls played in the open park rather than a bowling club using a proper bowling green? Or perhaps he/she is thinking of the Forest Hill Bowling Club, a short walk away off Wynell Road.