Well done John.
This is the birth which John found and that "cracks the case".
Births Mar qtr 1896
BATES Leah Renick Wakefield 9c 66
This is probably the birth of the second child.
Births Sep qtr 1897
RENICK James William Sculcoates 9d 185
With the first child maybe being called Leah Renick BATES was she illegitimate and the parents married after. Note that if Elizabeth was called CRAVEN (say) at birth then married a man called BATES then a second marriage would be using her married name but the children's birth certificate would still show her maiden name and there would be no mention of her first marriage name. Might be why the marriage records are elusive.
This could be a marriage of an Elizabeth BATES to Amos BATES but if Elizabeth had been born in 1873/74 then she would be very young. Legally possible but unlikely. Maybe she lied about her age in the censuses.
Marriages Dec qtr 1887
BATES Amos Wakefield 9c 84
CRAVEN Elizabeth Wakefield 9c 84
EXLEY Henry Wakefield 9c 84
HEBDEN Sophia Wakefield 9c 84
The 1891 census seems to show that Amos did marry Elizabeth who was (in 1891) age 21 and they had a daughter called Eva age 2.
In 1901 Amos is shown to be married and is living with his mother with his daughter Eva but without his wife. So I think Elizabeth married Amos then left him to live with RENICK but may have never married again as that would be bigamy - unless she waited until Amos died.