Hi
Could be off on a tangent here but is this a possible:
The 1901 census seems to show William as a visitor, working as a 'builder general labourer'. Other people in the census are a Cecilia Kenneth (58), Eliza Kenneth (19) and Charles Kenneth (15). It is showing he is still single, but that he was born in London, Westminster.
Is this him? Or a different William?
Otherwise the next time a William Race seems to appear is in the early electoral registers of the 1920's, however, the name appears alone.
There is also a William Race living at 43 Aston Road, Bromley who died 12 March 1941. Could this be your William? Informant was an Emily Akroyd.
Can't find a marriage, so possible he never married?
Cheers
Caryn