A hammock with tarp is a good idea for France during the summer. Especially if you take additional a ground sheet with you to use it on camping grounds, if there aren't trees. A walking pole you don't need. If there isn't a tree to attach the tarp, there is a hedge or lamp or whatever.
The ground is dusty and bright grey in sothern France, you should use bright grey clothing.
In most areas there are a lot of stones.
Such an air mat is a good idea:
https://www.decathlon.co.uk/trek-700-l-mattress-yellow-id_8493395.html
But only hammock and ground sheet for emergencies should work too.
For example this cheap and light poncho:
https://www.decathlon.co.uk/glenarm-hiking-poncho-green-id_8010059.html
I recommend to take tough lightweight aluminium pegs with you. You will probably get problems to get wooden pegs in the ground.
like this here:
https://www.decathlon.co.uk/ultralight-aluminium-camping-tent-pegs-id_2873930.html
If you do your trip during the summer, forget wood fires! You can get in real problems in France if you light a fire in a forest or somewhere else. It doesn't matter if wood, gas or spiritus.
At camping grounds you could use this gas stove for example:
https://www.alpkit.com/products/kraku
Big fitting gas bottles you can get in nearly every larger supermarket in France, smaller bottles at Decathlon, what you find every where.
(Dont use Optimus bottles with this stove.)
Think about eating only cold food at or from the bakers shops and super markets!
I usually do it there and do not cook.
Instant coffe works with cold or warm water from the tap.
Cheap camping grounds you find here:
http://camping-municipal.org/
But private owned 2 or 3 stars camping grounds are good and cheap as well. And they are every where.
The youth hostels are relatively cheap too, the standard is very different from town to town. But most I like. They are nice to meet young people and to visit towns, because they are located in towns or next to them.
https://www.fuaj.org/
Look at this thread here:
https://bushcraftuk.com/community/index.php?threads/french-maps-right-to-roam.151808/
If you want to buy a pot and a mug have a look at this:
https://www.tomshoo.com/p-y6549.html
In France I use during the summer the Snugpack jungle bag in coyote brown, because it blends perfectly in the nature in southern France, olive green would probably work too, but not as good as coyote brown.
This sleeping bag is perfect for French summer conditions. And it should work as a hammock cocoon / underquilt too, I guess, perhaps with a few modifications to attach it.
https://www.snugpak.com/outdoor/jungle-bag