Have been thinking this through again.
What about the hypothesis that the father of Ada Pay's children 'William James' was unable to marry her, because - perhaps - he was already married.
So she registers the children correctly under her own unmarried name of Pay, but 'legitimises' them by suggesting that her own name is Smith, her mother's maiden name.
This would explain why she then marries subsequently under her name Pay, but makes up her father's name as George Smith to match everything else.
So as you suggest, Rosie, we are looking for a William James who is a barman in the right sort of area for the births.