Sorry, I have been having internet problems

The only way of finding out what is on a birth certificate is by ordering it. I have in the past ordered one from the GRO and said that I do not know the reference, then you get a different form to complete. It does take longer for it to come as they will do a search.
I think if I was going to try this one all I would enter on the form would be the name of the child approx year of birth and give mothers christian name. If they can't find a birth for a Marie Saunders c1894 with mother Ernestine (I would not give Ernestine a surname) then they should refund the money if they can't find a match.
I am pleased that you have managed to get some more information from your cousins, it certainly sounds promising.
In 1891 Eugene is in Bishopsgate at 9 Crosby Square with wife Hannah and children Julia, George & Albert. Also a visitor Sarah Scott.
Ref RG12 piece 241 folio 35 page 7
In 1901 there is just Eugene and wife Hannah living Hammersmith
RG13 piece 50 folio 168 page 15
I think the Le fort family are certainly worth investigating further.
If Ernestine Saunders is previously Le Fort you may never find a change of name.