I thought that your idea was such a good one that I wanted to assist if I could so I spent an hour earlier looking up your chap.
His name was actually Charles Edmund Fawcett b. Bradford 11/3/1886. Son of John Robert Fawcett b. Leeds 1866 and Elizabeth Ann Clegg b. Gatehouse, Scotland 1863.
Baptised St Peter's Bradford 18/4/1886 (Became Bradford Cathedral 1919)
1901 Census: 44 Fieldhead Street, Listerhills.
1911 Census: 4 Primrose Terrace, Bradford. Lodger at his future wife's home. Ellen Williamson is 26 and a restaurant waitress. Charles is 25 and a Dyer's Labourer.
(Primrose Terrace no longer exists but 1911 Census return suggests it is just off White Abbey Road. Near to existing Primrose Street)
Charles had 4 brothers (Joseph Robert, Richard Clegg, William Frederick and Henry) and 2 sisters (Mary A and Isabella)
As Ady has already stated your man was not a Bradford Pal but his brother Joseph was.
Pte 18/176 Joseph Robert Fawcett was Killed in Action (KIA) on the first day of the Battle of the Somme (1/7/1916) serving with 18 Bn West Yorks (2nd Pals) Age 27. He has no known grave but is commemorated on the Thiepval Memorial in France. Additional information states “Husband of Rhoda Fawcett, of 19, Lower Lane, Wakefield Rd., Bradford, Yorks.”
Pte 341036 Richard Clegg Fawcett – 1st Bn Northumberland Fusiliers KIA 23/8/1918 and is buried at Douchy-Les-Ayette British Cemetery in France. (CWGC has his number as 341026. Medal Index Card 341036)
So it is true that Charles had two brothers killed in WW1 but not on the same day.
I couldn't confirm military service for William Frederick but did find his marriage to Amy Hardaker 26/12/1919 at St Mary’s, Laisterdyke. He died in Bradford 1972 Age 76.
The fourth brother Henry died in 1912 age just 11 years old.
Charles married Ellen Williamson in 1911 and as far as I can tell they had just the two children that you mention.
Mary A.P.L. b. Bradford 1912
Emily b 1915 Bradford. d. 1917 Age 1 Yr.
Charles died in Bradford 1952 age 65.
I hope this is helpful and is not too late for you. If you need anything else or more detail please get in touch.
Martin