Sure, and they'll be looking out for foreigners in suspicious white vansDrive Taking the van. Went through on the bike back in the mid 90's and never bought a vignette. Took my chance and got away with it but technology has moved on a tad. Ill sit down with Google and my sat nav to leave out toll and motorways and see what comes up.
I did it in a van: a full day from the French coast to the Alps stage. Nice drive. You could make your day up [timewise] there, and take the first three of your days at a more leisurely pace, taking more in. I know a wild-camp site just after that Alps stage, which is where you'd need to be on the 3rd night, up just beneath the southern end of the Jura, high in some vineyards with spectacular views. Its legal & free. The simple rule is its respected. Even camp fires are allowed. This would also give you much more time to navigate through the mountain stages.