If she couldn't write herself, it's more likely that the spelling "Faux" was not (necessarily) the official spelling or the spelling at her birth. The fact that she's listed as "Foulkes" on the birth certificate makes it more likely that there were lots of variations in the name over the years.
The coverage of the Consular records is not complete; it was not required to register children born overseas. Another thing to check would be whether David had several of the children baptised in a group upon returning with his new wife to Llangystenyn. It's possible that her birth was registered in the French records, perhaps in the same area where her mother died, but if the family were moving around with the rail a lot it might be hard to trace (Calvados archives are here:
http://archives.numerisees.calvados.fr/ but require a subscription to access).
Also interesting: in 1861 there's this girl living in Llysfaen:
RG09/4358 5/4
(address Pendyffryn)
William Foulkes, 68, Ag Lab, b. Bett?? Denbighshire
Mary, 67, wife, b. Llanfait ditto
Elizabeth, 4, granddaughter b. France
(the French records have the parents of David down as being William and Mary)
and at
RG09/4358 12/18
(address Fronfela)
Peter Hughes, 36, farmer 27 acres b. Llanfaen
Jane, wife, 34, b. Llan?ian, Denbighshire
William, son, 11, b. same
Caroline Foulkes, nephew (clearly written but age in female column!), 9, b. Liverpool
ETA: checked the Consular records, none of the Foulkes children were registered, nor was the death of David's first wife. Probably as they were travelling around between France and Italy etc it wasn't convenient for them.
