Hi Jesse,
I've gone back and checked through the mid-1950s; I feel comfortable saying that "F E Bowden" as such disappeared from the Exeter directory after being listed for a final time in the 1942 edition. I can think of many alternatives to explain the fact pattern, but perhaps the simplest is that the pipe dates from before that time, very likely from before the start of the war.
If it's of any interest, the Bowdens were tobacconists in Exeter for many years. Fred's business was originally established by an uncle named Edward Albert Bowden (initially partnering with one of his in-laws named Anning) sometime in the late 1870s; by 1881 Anning was gone, and the business was in (and stayed in) the hands of the Bowdens for over sixty years.
Edward Albert Bowden (1849-1916) had an older brother named Harry Francis (1846-1898). About seven years before Edward Albert died he turned the business over to one of his deceased brother's sons, Francis Leopold Bowden (1882-1917). Francis took on an old and valued assistant of Edward Albert's as his partner. This man was George Farrant Chalk, and the resulting enterprise was restyled Bowden & Chalk. Francis died in the Great War while serving in France as a private in the 8th Middlesex Regiment, and two years later his older brother took over the business. This brother, Frederick Edward Bowden, is the one whose name appears on the Barling as "FE Bowden". Fred had spent a good part of his life working as a banker (god help him), retiring in April of 1919 with 25 years of service with the Exeter Bank Branch of the National Provincial Bank. Tucking the obligatory gold watch chain into his pocket, Fred began the next phase of his career. At age 41, he took sole ownership of Bowden & Chalk, eventually renaming it FE Bowden about 1929.
Jon