I'm coming here by chance and this is my first post. I'll introduce myself later.
As a French rider, I can give you some input on the French areas mentioned in your post.
The Loire Valley may be a flat especially if you follow the river. But you can reach it easily from Paris by train with your bike. Along the river you have many trains TER where you can travel with your bike.
Provence may fit your specifications better. In Provence there is an area called "Luberon" located East of the city "Avignon" which may be very nice with your program.
And the weather maybe nicer in the Southern part of France then in the Loire Valley.
There are many great places for riding in France and I would recommend to think also in terms of logistics:
Will you have your bike or do you want to rent one?
Will you come by plane to your final destination (if so it may not be necessary to book take the train)
If you want to check the topography, you can watch online free topographic maps on the website
geoportail dot gouv dot fr (I'm not allowed to post links...)
The "canal des deux mers" is nice (this is part of my summer 2022 trip). But if you follow the route mentioned above, the ride may lack of challenge for you @
ridethetown
It will be pretty flat.
You will find more challenge if you leave the canal and reach the more hilly areas on each part of the plain of the canal, more particularly between Toulouse and the Mediterranean sea.
In this area, there is a nice touristic place called "pays cathare" around Carcassonne with nice medieval castles (ruins sometimes) on the top of hills. It's a good aim for a one day ride!
Be careful with the railway national company SNCF that claims facilitating transportation of bikes... Iike in the link above. The reality is a bit different.
But, if you take regional trainlines named "TER" you will have in every train free space for bikes. And you don't need to fold your bike. This is the most convenient way.
In trains where there is no specific space for bikes, mostly high speed train "TGV" then you will have to fold your bike. It means that you need to fold the handlebar, pedals, wheels of course and to store the bike in an appropriate bag (or big garbage bag held with scotch that wou can throw away after the train journey).
In general in France, you will find a good network of quiet roads and many tracks in the countryside.