Yes I think one possibility is that Arthur was not born to William and Emma, but was perhaps a grandchild, illegitimate or otherwise. Have you accounted for all their children?Alternatively they might just have been one of those families which was not consistently compliant with registration requirements.
How many of their 10 children did they register? Is Arthur the only one missing, or were they casual about it generally?
I can see these children:William, Mar qtr 1891Rosie, Dec qtr 1893George, Mar qtr 1896Annie, Sep qtr 1897Emma Lydia, Sep qtr 1899Edward Thomas, Mar qtr 1902Frederick, baptised 23 March 1904
Have you found a birth reg for Frederick then (1904)? You say you have them for all known children except Arthur.For me the most likely explanation is that Arthur Douglas/Douglas Arthur was a child of this family, born on 1 May 1908 but not registered for whatever reason (probably of the “didn’t get round to it” variety). Later he met and married Lenora, altering his age so as to appear older than her.