Ah, here we go!
In Belgium there are "marriage supplements" (Huwelijksbijlagen) these are documents supplied at the time of marriage and the ones for Antwerp are online!
https://www.familysearch.org/ark:/61903/3:1:33S7-9THH-9GDV?i=78&cc=2138481&cat=57143These are what William provided at the time of his marriage - proof of his baptism and that his parents were deceased.
- Death certificate for William Cotgrave/Cosgrave Bethell, 1848 aged 29. (A printer, died in Liverpool)
- Death certificate for Mary Parkinson, widow of Thomas Parkinson, died at the Liverpool workhouse in 1861 (5th May so might just have made it onto 1861 census?) aged 49.
- Copy of the Catholic baptism in Liverpool in 1842 as found above.
(Interspersed with translations / certificate of these records)
There are also some records for his wife's side of the family. This is a summary and there is more to be found in the details of those records I'm sure.
https://www.familysearch.org/ark:/61903/1:1:N5G5-HMR - here is the baptism of William Sr in 1818
I think this may be William with mother Mary in 1851, although hard to be 100% sure.
https://www.familysearch.org/ark:/61903/1:1:SGPP-VPNThere is an 1849 birth for a Maria Parkinson in Liverpool, mother McCoy
She may have died in 1860 (aged 11).
No sign of a marriage for Mary/Maria McCoy to Thomas Parkinson.
Records for Liverpool Workhouse may help?