Everything points to Messrs Braithwaite being the firm of Albert Braithwaite, born 1868, Leeds, the son of William Braithwaite, plumber and coppersmith of Swinegate Leeds.
Note the last sentence of HT’s previous post, a reference to Lord Mayor. Albert Braithwaite was Lord Mayor of Leeds 1920-1921. There are numerous items from the mid-1920s about his involvement in a legal case concerning disputes over him participating in assigning building contracts as an Alderman of the Leeds Council when he was himself in the trade.
Having dug around in a few sources about the family I have found no hint that Walter Samuel Braithwaite was a relative.
added: his 1945 obituary is here for anyone with BNA access:
5 February 1946: Yorkshire Post and Leeds Intelligence
https://www.britishnewspaperarchive.co.uk/viewer/bl/0000687/19460205/169/0006