Hi Janer.
Re, Bathsheba, I did originally research her and her link to the Smith's of Birkenhead, but it was in the mistaken belief that our family from Liverpool was in someway related to Frederick Smith, Lord Birkenhead.
Even now my elder Bro some 85yrs old still insists there is a connection somewhere along the line, maybe through marriage of one of his children ? don't know, can't find any, the only link to that side of the water is my GGGrandad who married a girl from Neston, and lived in Higher Bebington, Cheshire.
Sorry I can't be of any more help to you, good luck in your research.
Kind regards Scouse.