Chains don't help much on summer tires. They help though a little bit...if you put chains on all four tires.
Driving to the french alps with summer tires is the dumbest thing someone can do. Not only do you put YOUR life at risk, you also put others at risk.
I met such a clever person last year in Austria, a family father from Liverpool with a Range Rover. He not only rear ended a friend of mine, he didn't "get it" when the police tried to explain to him why driving in winter Austria without winter tires and specifically on that street is ILLEGAL.
In France, some mountain roads request winter tires, it is the law. For all skiing regions, winter tires are highly recommended and trust me, if something happens, you won't find anybody from french authoroties having mercy with you.

Why don't you just fly over to France and rent a car with winter tires ? Lots of lads from Britain do that.