ninja'd by Carl S.
You are too pessimistic.
I can tell you this was hotly debated and wargamed altogether, and early August wasn't
that optimistic (I've heard some scenarios had a small corner of Southern France holding until September).
It seems the key issue often underestimated is German forces slowing down mid June (around OTL armistice) because they were overstretched, just like the Wallies in the fall of 1944 (except North > South instead of West > East, but France is an pretty symetrical hegaxon).
That's overstretching and slowing down has been discussed a lot, some readers believe it could have happened, others not. Admittedly, it is a critical point in the FTL scenario.
The French government reorganize around Toulouse and Marseille. The Rhone valley is a bottleneck stuck between Massif Central and Alpes, giving a major advantage to defense.
Bordeaux isn't really relevant: it is used as long as possible but given up early July. Most of the Atlantic coast isn't really useful, the main effort is around Marseille and Toulon and the Mediterranean, so Germans are left overstretching on South-West France, the most of the defence being around Toulouse.
Having grown there, I can tell you the land stretching between Bordeaux and the Spanish border is mostly empty space, except if your panzer engines run on sand and pine trees.
For example, there is one squadron left to defend South-West airspace: a dozen of Bloch MB-152s, reinforced by whatever MB-155s they can get their hands on (Bloch = Dassault = Mérignac = near Bordeaux). I wrote their adventures ten years ago.
An example of a the possible German overstretch is Me-109s being limited by range, as usual, and it is not a matter from leaping to a new airstrip southward: they are deliberately destroyed by retreating French troops). From a brief moment late June and early July, escort has to go to the longer ranged Me-110s, with their usual caveats. Even a MS-406 can score against a 110.
Also, from early June French troops and aviation fought much harder, lessons had been learned the hard way, but too late.
(I made a mistake: I thought the fight stopped on August 15 but it actually stops on August 9).