Weather IN FALL in Burgundy, Provence, Cote D'Zur

I prefer late September to October, and hopefully you will be there when the grape vine leaves turn gold and bright orange/red. This is why the region is called the Côte-d´Or.

For the best chance of catching fall colors in Burgundy vineyards, you should be there around mid to late October.
So I would also pick "late September to mid-October" and I would start with Côte d'Azur (land in Nice if possible), then head west to Provence and north to Burgundy.
The weather would probably be better the other way around, but fall colors in Burgundy are incredible. I'd happily forgo outdoor dining to enjoy them.

If you want to beach and swim, you are better off arriving in the middle of September in the cote d'azur. (no guarantees, however). The beach clubs start to close down from mid to late September, depending on the beach. The water is warm, however. Early October can still be good but it can be dicey the further you go into the month (but hey, I dined on the beach in late November this past fall - but in a light jacket)

Then head to western Provence and end in the fall colors of Bordeaux.

As a former resident of Bourgogne for several years, autumn can be quite nice on a dry day. My advice would be to move from south to north on your sojourn, unless you're starting in late October. I'm not sure how much you can take in of "the way of life" with just a few weeks, but you should be able to get a very, very, general and superficial sense of whatever areas you choose to visit in that short time.