I am not totally sure what you mean by "the Switzerland fee". I Guess you mean an Autobahn Vignette. Perhaps you could come back and clarify.
An Autobahn Vignette is a toll for using Autobahns=Autoroutes=Autostrada=Motorways. It cost CHF 40 for one calendar year. It comes in the form of a sticker you attach to the windscreen and looks like this.
You only need to buy one if you are going to drive on a Swiss Autoroute/Autobahn/Autostrada. If you keep off these roads, and very importantly, do not cross the border on a Autoroute/Autobahn/Autostrada then you do not need to buy one.
Autoroutes are clearly marked on road maps and on signs, they are easy to spot.
If you are very lucky the hire car will already have been driven in Switzerland this year and will already have a Vignette.
Edit: I just checked the route on www.viamichelin.com
Shortest route: 43 Km, 0h40, €6,90 French autoroute tolls and this route does include a Swiss Autoroute.
Avoiding the Swiss Autoroute (and avoiding the CHF 40 Vignette): 47 Km, 0h57, €6,90 French Autoroute tolls.