I'm aware of two choices, both involving around 12 hours elapsed time: (1) drive to Genoa, then take three trains to Aix TGV (depart Genoa 12:55pm); (2) drive to Rome FCO, easyJet to Nice (depart FCO 3:55pm), rent car to Aix. Is there a better alternative, and what about the drive on A1 to Rome/FCO mid-day, mid-week?
Why not drive the whole way? You may even shave a few hours off your time or you could use them for a nice lunch stop along the Med. It's a gorgous drive. Bring plenty of change and small bills though as there seems to be a toll booth every few miles. I was recently driving on the A1 in the morning and mid day and neither was any worse than it is here in Atlanta. Mid day of course was less crowded. About a year ago, I almost missed a flight because of a traffic accident that shut the whole thing down.
I have researched going the other direction and there is no good way to do it. Four scenic trains either along the coast OR through Switzerland in the twelve hours you mentioned. I wouldn't try to do it in one day because if one is off your itinerary is shot and chances are there won't be another late train. Another option is the overnight train which saves you a nights hotel but I don't know if they run that direction. There are a lot of cheap flights to Paris and then the high speed train would probably be less time overall. Call Eurorail and pick their brain for suggested routes (whether you book with them or not). RD