I'll never understand all this with insurance in the UK not covering Europe. My insurance in Luxembourg covers me everywhere throughout Europe (and globally depending on how ambitious I'm feeling for a drive!)
As for your plan Jared, any specific area of France you wanted to go to?
As you'll be either P&Oing it or Channel tunnel, you could just as easily go through Belgium, Luxembourg (where I'm based - for a quick and cheaper fill-up) and Nurburgring. The issue with France is the toll roads. For example, you'll usually be paying around £70+ to drive down to Nice/Cannes and then you have the same outlay on the way up.
Another option would be to travel through Germany and Switzerland to Italy - bit of Stelvio Pass and Milan... all depends how much driving you intend to do in a day..