Upvote:3
I guess you must have solved your problem since but the only solution for you would have been not to go through a car rental agency but through an individual in this situation.
I know numbers of my colleagues who have their papers in order rent through websites like "leboncoin" or "PAP" and the owner of the car which is an individual would just require identification papers but not necessarly a drivers licence. And for paying cash would have worked with an additional security deposit in cash as well.