Hi there,
Paris civil birth records from that period are among the records destroyed during the siege of Paris in 1870/1 (there are some reconstructed records but they are considered to be only maybe a third of the total).
However, as she was born to British parents, you also have potentially access to overseas births and baptisms - e.g. her parents may have registered her with the British consulate, and she also may have been baptised in an Anglican church in Paris, or baptised when the family returned. There are various different sources:
https://www.familysearch.org/wiki/en/British_Births,_Marriages_and_Deaths_OverseasIt was not required to register an overseas birth so the coverage is unfortunately quite spotty.
Trying to put things in a timeline - would be very helpful if we could find some reference other than her census records confirming where and why these families were in France!
1851 - with Alphonse Fitzjohn in Chelsea
Mar 1851 - registration of Harry Fitzjohn (shown on census) but with mother's maiden name blank on GRO
1853 - birth of Mary Ann Elizabeth Fitzjohn / Fitz John in Chelsea - mother's maiden name Prince. Appears as Mary Webb in 1861 census. No later sign of Alphonse.
1855/6 - birth of Louisa (varies a little but this is consistent with the earliest census records ) - middle name Adelphine, later gives her father as James Austin Webb and is fairly consistent about using Webb.
1858 - birth of John Richard Webb, mmn Prince, in Sussex (on 1861 as Richard Webb)
So we are looking for traces of this family (Fitzjohn, Webb, Prince, or

) in Paris between 1853 and 1858
http://archives.paris.fr/s/5/etat-civil-reconstitue/? - reconstructed records of Paris show:
Charles Francois Webb, born 26 April 1854
Henriette Webb, born 18 February 1856
Anna Webb, born 7th August 1856
Florence Eugenie Webb, born 11 August 1855 (11e Arrondissment) - there is also a death under this name.
Madeline Antoinette Webb, born 14 April 1858 (10e) - there is also a death under this name.
There are no Fitzjohn births in this record set.
Can you find any other traces of Alphonse? Do you have info on James Austin Webb's whereabouts before 1861? One slightly wild suggestion - with the even spacing of the children, is it possible that "Alphonse Fitzjohn" was simply an alias for James Austin Webb?