Even combining all Daniel and Margaret Moore entries in the irish census 1911 I could not find them living together, which suggests to me they married post 1911. Wrong see later!
One Daniel Moore who might be related in 1918 is living as
grocer and confectioner, 225 Ligoniel Road, Belfast see
www.lennonwylie.co.uk Then I realised I had not looked for a Maggie Moore and so found a missing match on the 1911 irish census
http://www.census.nationalarchives.ie 118 Ligoniel Road (part of) (Shankill, Antrim)
a Daniel Moore aged 27 so b about 1884, runs a
newsagent with his wife Maggie 27 so b about 1884 they have been married 6 years and have three children the family is catholic which might help for church records in central or outer Birmingham registration areas?
Looking at the 1901 census see
http://www.census.nationalarchives.ie at 14 River Street (Shankhill, Antrim) the entry looks like this could be the young aged 16 Daniel, with his family inc his father Daniel aged 45 and and Maryan his mother
Turning to Thomas Moore born Belfast 1914 a Thomas with those details is on two trees on Genes Reunited, the tree owners are an ‘Oliver’ and ‘Peter’. To see either tree you would have to join and then ask their individual permission to see their trees Also on this website looking for Thomas Moore’s death it suggests he was born 1st May 1914 You could get a copy of his birth certificate from GRONI Also on this website an ‘Agnes’ has a tree that includes a Daniel Moore b Belfast 1890, but I think I would discard this as 6 years out
so I think there is hope in finding more information, the easiest way being Genes Reunited
I also did wonder about WW2, he might have served in armed services etc, so he might have war medals reserved for him see National Archives Kew. Have you tried Birmingham street directories some might be on line? His death registration Birmingham could mean a cluster of locations around Birmingham
Good luck onwards JimG