No Arthur Frederick Smith in the inter-war Beckenham directories that I've spot checked, but some houses had more than one family and only one is named. There was an Arthur Smith without a middle name in Blandford Road, but he had gone by 1939. I don't have a sub for the 1921 census, but if you have the address there (or from elsewhere) and will tell us here or send me a PM, I'll check further, including the 1939 Electoral Roll.
I can't see him in the Service Register pages of the 1945, 1948 or 1949 ERs. I don't have these pages for other postwar years (if they were printed).
He lived with his family at 7 Lennard Road, Beckenham in 1921. But I know he wasn't there in 1939. It was just his father, step mother and half sister still living there. And his step mother's mother.
