Hi and welcome to the forum.

The best resource online for questions about what rights foreigners do or don't have in France is the Service Public website here: Etrangers en France - Service-public.fr

Now, just in case you can't find the answer to your question on that site I can tell you my experience. It depends a bit on your local prefecture (and you might ask at the prefecture de police for the 15th). I was supposed to show a carte de séjour in order to register my car when I arrived in France (I'm in a departement in the Ile de France region). I weaseled out of it, and it wasn't that difficult to do. (I dare not tell you how I did it, since I'm not sure my technique works any more anyhow. It wasn't supposed to work when I did it... but I did have a recippissée for a carte de séjour at the time, and I knew the prefecture didn't answer their phones after a certain time in the afternoon. Sometimes, if you throw yourself on the mercy of a fonctionnaire, you can work miracles. )

I don't know that you need a carte de séjour for the insurance. I would ask at a number of insurance agencies - or ask the conseiller at your French bank. (They sell insurance, too, and are usually quite eager to find a way...) If you can register the car, I would think you could find someone to insure it for you.
Cheers,
Bev