Alex,
I spent a good while looking for her on C1881, but without success.
Although the absence of her 1881 record is a "loose end", it has to be weighed against the positive bits of info on her...
e.g. her living with her mother ( or her husband's mother in law, to be specific) on C1901.
A useful strategy where say there is doubt about which of more than one candidate, is the right one, is to track, back and forward, those candidates, hopefully eliminating the wrong ones.
In the case of Elizabeth Proudfoot... C1901 says born in Benwell,c1851. OK so find this Elizabeth born in Benwell c1851, who had similar-named children also born in Benwell. And when/if you find these, then you can reject the " born Carlisle/Penrith " scenario.
To be sure, about Frances Proudfoot, you might buy her birth cert, and two marriage certs, looking for all three certs to reveal her father's name, and one (birth) to reveal her mother's name. and to reveal she was a widow at time of second marriage., etc
Taking into account the fact that surname Proudfoot is not common, and the geographic compatabily, I personally would not buy the certs, unless the Lotto came up for me.
Michael Dixon.