My autoroute reckon it about 475miles Calais to Lyon. Which would seem about right as I estimate 10 hours for the section in the truck.
I don’t know what’s cheaper, but as for route I normally go Calais, Reims, Troyes, Dijon, Lyon. More experienced driver might have a better route.
Don’t know how long you are thinking of taking, especially on the way back with a sick relative in the car, but there are plenty of hotels on the way, either on the services or very near. I can get you details of the ones we use.