Bike touring France

There is a possibility you will have terribly hot weather in all your French cities with the possible exception of Chamonix (which I haven't visited). That's what I would fear, based on my 2017 trip to eastern France. Annecy is not up in the mountains, though near them, and it was very hot last June. I am not remotely athletic enough to contemplate traveling primarily by bicycle, but the potential weather would give me pause.

I have no idea what sort of terrain you would face, connecting those dots, but I'd like to point out that the trains are great, and you could rent bikes locally to enjoy suitable parts of Provence, to circumnavigate Lake Annecy, etc.

No craziness that I can see here.
Crossing the French-Swiss border, you will not be stopped. There is no passport check as both countries are in the Schengen Area, and customs won't be interested in the amount of goods on a bike. You probably won't have to slow down.
Chamonix or other Alpine places will be strenuous, but plenty do it. They even cycle across the high Alpine passes.

Campsites - research and pre-book. They will have showers.

You can get Swiss cycling route maps online: https://map.schweizmobil.ch/?lang=en
Click "Cycling in Switzerland" and zoom to your required area.

We are avid cyclists and drove from Lautrrbrunnen to Geneva. We did remark that we would not want to be cycling not only due to the tough uphills but the hairpin turns on the steep downhills. However you should probably check out details and possible routes.

We did cycle in Provence and it was wonderful.

Is it crazy to imagine biking in Chamonix or anywhere near the Alps for that matter?

Nope. Assuming you've done hills in CA or the West Coast then you'll find the Alps generally easier. The overall climb may be longer but in general the grade is not as steep. Several years back I was on a biking trip starting from Geneva and we went down to Grenoble. Went up several cols including Alpe d'Huez, Columbier, etc. French drivers are pretty polite on the small mountain roads, they don't honk to hurry you past the turn or try to run you off the road.

If you have a local cycling club you can see if there are other members that have done a similar tour and get advice from them. One blog about the subject is http://piaw.blogspot.com/ and he also sells a book with advice. There are also adventure bicycling tour companies that offer various levels of support from a fully-supported tour (you ride,they carry the luggage) to ones that perhaps just help you set up the tour but you do it on your own.

On my trip there were several people that had done fully supported tours in the past so they knew the area, some good rides, etc. We rented a van and would spend 2-3 days at hotels in towns along the route. Everyone took a turn driving the van between hotels with all the bags, so on the road all we carried was the ordinary stuff for a day ride, much lighter and easier than carrying all your gear.

To me, the 2-3 day stop was ideal because if someone was tired or not feeling up to it they could do a shorter ride or just take the day off. And there are lots of interesting rides in the vicinity of many of the towns in the Alps. Others might want to push it another 100K farther each day like you are suggesting, its up to you.

there is a possibility you will have terribly hot weather in all your French cities

part of the fun ;-) Our trip was in 2003, the year of the heat wave with people suffering in Paris. We were riding up Alp passes in 90-100F days. If you train for it you can do it, just keep hydrated. The water at all the public fountains is drinkable unless marked otherwise