We have found we prefer to use public transportation and bicycles to explore Europe rather than renting a car. We are hoping to have our own bikes this year (folding bike - Bike Friday) and enjoy renting an apartment/gite for a week at a time and explore an area by bike. Anyone with experience traveling in France by train and bike would be appreciated.
I'm sure you will get information specifically intended for cyclists that will give you a heads up on which roads are safe for cyclists and which may not be recommended. I say this because we have noticed that in France a surprisingly large number of the scenic roads are very narrow with blind curves and no bike lanes or even shoulder.
Your plan sounds very appealing. When would you go?
I think the choice of folding bikes means that you can bring them on any train, including the TGV, not just those where cycles are permitted. And, if you pack appropriately, you can just ride from the station to your gite.
You've probably thought of all this, but here are a few issues that occur to me.
These bikes look pretty cool, but I am not familiar with them. You probably want something that is going to be OK on dirt towpaths and cobblestones.
Michelin maps are helpful, as are IGN topo maps. There's also an out-of-print Lonely Planet book with French cycling routes. Not day trips, but you could still crib ideas (or consider an overnight trip.)
There is no substitute for local knowledge, though. Could you befriend--or offer to pay--a local bike shop for route advice?
Finally, consider that on rainy days a base near some sights and a train station might be better than bucolic isolation.
Have a really great trip!