This is a terrible idea :)
You can't travel on french highways with a 50cc scooter, it's forbidden by law. If you do that, you will likely get arrested by the police and your scooter will be seized.
You can decide to avoid the highways, so why not?
Well, a scooter's engine can only take so much drive before it overheats. Basically, if you travel non-stop for 3-4 hours, you may seriously damage it. I know because I traveled from Paris to Picardy on a 125cc (about 1h drive), and my engine suffered from it. This is the main reason why people who want to travel buy a motorcycle instead of a scooter.
Finally, you should consider the amount of gas and the speed at which the gas goes away in a scooter. You may have to stop at every gas station to refill your scooter, which wouldn't be very convenient.
All in all, get the train, rent a car or buy a motorcycle =)