Just looked at the IGI & there are two baptisms attributed to a James & Ann Gawler;
Ann 13/5/1819 St. Mary Lambeth. Extracted
Maria 8/1/1830 ditto. Batch no. 1016582
A marriage would have taken place prior to 1819 which narrows it down a bit. As James was a Parish Clerk it would make sense that family events took place in the same church, most likely St. Mary's?
FreeBMD gives a death for Ann Gawler 1844 in Lambeth
FreeBMD has the marriage of Caroline Catherine Lydia Miller to James Gawler, Lambeth (4 393) Q3 1849
Another thought might be that her mother, Martha, remarried. I had a look at the 1861, 1871 & 1881 censuses & it's very difficult to read. I tried to find them (Martha & Caroline ) in 1841 but no luck

Tried to find the death of Martha but proves difficult as we can't decipher her surname. Caroline & the family are in West Ham, Essex in 1881 so I presume that's where Martha dies.

Saw that there's a 36 year age gap between James & Caroline

jane