Alice was 39 at the date of the marriage and 42 when she died 3 years later, so almost certain to be the same Alice not a second wife of the same christian name.[this happens on my tree and has caused some confusion to other researchers who have not looked at the PRs, only the census.]
Have you considered that Robert and Alice were not living together, maybe even living with other partners, ie; the speedy marriage of Robert after Alices death. She[Alice] would still be his wife and the record would have to state that, and also as husband, the father. With his family being so involved with the church, maybe even writing up the records they would take the opportunity to make a note to try and distance him from the child.
Maybe it would be possible to find where they are living, but not being census years the search becomes more difficult.
If you have access to the PRs[or copy of] then you most probably will be able to see who wrote them.
