The directories give some information, though, as date on them is publication date, year referred to can be the year before. Details of the Barrel from them are below:
1829 Thomas Castle
1833 John Gayley
1839- 1841 Benjamin Bridgewater
1845-1862 Mrs Mary Bridgewater
1867-1884 John Yates Bridgewater
1888-1890 Mrs Eleanor Fanny Green
1892 Henry Watkins
1895-1899 William Butler
1900-1903 Arthur Wade Edge
But name seems to have changed to the Crown, and in 1904:
1904-1905 Charles White
1908 – 1915 George William Ansell
1921 – 1924 Sidney Samuel Fumpston.
1932-1933 Mrs Dorothy Maud Fumpston
1936-1950 Frederick Thomas Stansbie
1955 Harold William Rogers
1956 Richard William Fisher
No landlords are listed in directories after 1956, but pub Is listed till the 1961 edition, disappearing from the 1962 edition.
In 1833 Benjamin Bridgewater is listed as a victualler at 44 Birchall St
And in 1829 a Benjamin Bridgewater is listed as a gun & gunlock manufacturer at 57 Coleshill St.
Below is a map from 1880s showing the position of the Barrel inn on the corner of Bath St.