Here's something curious.
Thomas Cheyne Bee, aged 60, died in Dalkeith in 1957. If his age is correct he was born 1896/1897 but there is no birth of a Thomas Cheyne Bee at any time, and there is no birth of a Thomas Bee in 1896/1897.
However there is a birth in Dalkeith as Thomas Bee Cheyne in 1897, mmn not in the births index, which might imply that he was illegitimate.
Reply #5 above lists members of the household of Thomas Bee and Margaret Cheyne in Dalkeith in 1901
Thomas Bee 44 Brewers Cellarman
Margaret C Bee 22 daughter
Mary C Bee 15 daughter
Hugh C Bee 11 son
Agnes C Bee 7 daughter
But there are two more members of the household not listed in that post
Thomas B Cheyne 3, grandson
Robert Cheyne, 20, nephew
So if you abandon the idea that your Thomas Bee's mother's given name was Myrtle, and that his middle name was Andrew, Thomas Bee Cheyne, later known as Thomas Cheyne Bee, could be your elusive Thomas Bee.
However ....
There's a 23-year-old Thomas Bee in Dalkeith in the 1921 census, and there's a Mary Bee, 19, in the same household. Thomas Bee married Mary Reid in Edinburgh in 1920. Mary Haldane Reid, other surname Bee, died in Edinburgh in 1938, aged 36, and there are 9 births of Bees with mmn Reid in Dalkeith between 1923 and 1936. So if this is Thomas Bee Cheyne, he's rather too busy in Dalkeith to have married Dorothy Rouse in Farnham 1925 and fathered 5 children with her there between 1923 and 1938.
Inconveniently, however, Thomas Bee Cheyne married Isobel Dalgetty in Edinburgh in 1915. I have not found a record of the death of Is*bel* Dalg*t* with other surname either Bee or Cheyne, or any record of children of this marriage.
You would need to look at both marriage certificates to see who the grooms were.