It looks like James was a naughty boy....
Baptised 26 May 1816 Banbury St Mary was Elizabeth Izzard, d/o Jemima Izzard and James Wheeler, blacksmith of Newland. (James and Charlotte were married in the same church seven months earlier).
My understanding is that James would have needed to be present at the baptism to be names as the father in the register. I’m sure someone much more knowledgeable can clarify this?
Are you not thinking of a change in the law (1870's?) about including father's name on birth certificate of a child whose parents weren't married? Prior to then a single mother could name a father when registering her child's birth. After the law change, an unmarried father had to be present at the birth registration and give permission for his name to be included on the birth certificate.
When Elizabeth Izzard was baptised in 1816, poor law relief was the responsibility of the parish. Office holders of the parish would do what they could to find out name of father of a child of a single mother so that he could be made to pay for upkeep of the child + cost of the mother's "lying-in" if she was admitted to the parish workhouse for the birth & aftercare. A curate might write "reputed " or a similar word next to father's name in baptism register, indicating that the baby's mother or someone had given information that he was the father but the man hadn't admitted responsibility.
One of my ancestors was taken to court for maintenance of a baby born in 1823. He was named as father in the baptism register and 7 years later in the burial register when the little boy died. My ancestor was unlikely to have attended his son's baptism as the baptism was at the Anglican church and he belonged to a locally well-known Catholic family.
