Hi if you go to this site: www.familysearch.org and on the left hand table click on IGI data only (2nd one down).
For the forename enter * and on the surname enter Battersby - then select your region (I did the British Isles) you can pull out every single Batterby in the IGI.
Scrolling down - they were all over Britain but mainly in Lancashire and Yorkshire so I don't think the origins of the name would have much to do with Battersea in the south of England - but- who knows! If you go to this website:http://www.spatial-literacy.org/UCLnames/Surnames.aspx You can search on the origins and distribution of a name. The search results show they were actually from Lancashire and the name means they came from 'the end of a settlement', the demographic data shows this to be Leigh in Lancashire.