Following the previous posts from various helpful contributors, part of which concerned a putatative Jane Rawes, who may or may not have been the first wife of my 2xgrandfather John Bateman, I have been able to confirm that Jane Rawes was indeed the very person.
The breakthrough turned out to be two specific details from a proper copy of their marriage certificate (19-04-1882). Jane Rawe's father is given as William Rowlandson, and one of the witnesses was an Agnes Brownrigg.
Further digging and cross-correlation revealed that Jane's mother (Agnes Rawes) was single when Jane was born in 1860 at Dobsole (Dob's Hole) Lambrigg, but married William Rowlandson very shortly after (perhaps within weeks). In subsequent records (eg 1861 and 1871 censuses), Jane is then listed as Jane Rowlandson. So, no wonder it was difficult to find her in the years between her birth and marriage.
As to Agnes Brownrigg, she turned out to be Jane's first cousin - the daughter of Isaac & Isabella (nee Rowlandson) Brownrigg. Isabella being the sister of William Rowlandson. At the time of Jane's wedding she and Agnes Brownrigg were living on adjacent farms (Whin How & Deep Slack, Whinfell) - both run by the Rowlandson family.
Many thanks again to those who first provided hints and possibilities for a Jane Rawes. Without those kind contributions, I wouldn't have had a starting point, other than family memory that John Bateman was indeed twice married. Poor Jane died at 23, a year or so after her marriage.