I expect others have done the same thing, but I've just gone through a list of all known French car makers, concentrating on marques which were built in the period 1945-1960. I have discounted all major manufacturers, and any makes which were powered by rear engine, or engines provided by major makers. The puzzle car does not exist!
It's not-
Rosier
Salmson
Arista
DB
Barbier
Facel Vega
Amilcar
Rosengart
Gregoire
Chenard Walcker
I have discounted luxury marques like Delahaye and Delage - the puzzle car looks too small.
I have been through the entry lists for Montlhery in the appropriate period, and nothing fits! Surely it's not a rebodied Lambert ?