I was promoted to Expert Auto Puzzler before I had a chance to finish this on the Rookie level - so I'll try to finish it now: I believe the correct model designation for this car is the Bernardet 3828, pictured at the Salon Auto in Paris in 1950.
Edit: Actually, let me take that back. The web page where I found that designation is the only place where I can find anything resembling a name for this car - and it clearly is not '3828'. That is merely the product number of the old photography picturing the car that you can order from that page... a stupid mistake to make! So, I'll just stick with what I've said: It is the 1950 Bernardet "trois de front" ("three up front") prototype, so called because its seating arrangement, similar to that of the Matra Simca Bagheera. It had a 750 cc four-cylinder engine, and several orders were taken, but Bernardet's financial problems shut down the marque's car-making division, making them focus instead on scooters alone.
