newislander,

May, June July are the best month to go to France. Temperature are very nice and you should not have much rain at all. At this time of the year, rain is mostly from summer thunderstom so they don't last, so I would not plan on biking on rainy days much.

For the list you gave, I would remove all the items for cold weather:
Beanie, scarf (maybe a Buff, but that's it), rain poncho (even in South East Asia during the moonsoon I didn't use mine!), down jacket, thermal long john (as long as you have a decent sleeping bag, you will be fine, temperature won't go below 12C at night,). wool sleeved top, wool sucks.

Goretex pants and jacket are recommended only if you REALLY have to bike when it is raining, but seriously you will be better off waiting a couple hours. Otherwise, you should use only paclite Goretex as it is very lightweight. The best brand is Gore Bike Wear as their clothing are made for cycling (stretching knee, extendable back etc - not cheap though. If you take a goretex jacket, don't take a wind shell. It shouldbe one or the other, and you will only use them at night around the camp if the wind is a little chilly. I never heard of rainproof fleece, but this would be only for around the camp at night or in the morning, so you should choose one or the other.

As for how many you should take of each: socks and underwear: 2 to 4. Lycra: 2, biking t-shirt: 2 to 3. Everything elasem, you should take only one of each.

If you really want to bike under the rain, it will not be too cold especaiially in June and July, even in the North.

Nice route, enjoy!