Not taken my bike across the pond, just a few European trips and unless going to 'cycle' only... like a training camp; a good way is to re-use a shipping box from a retailer for the initial trip... then have a Seasucker bike rack (
https://www.seasucker.com/collections/bike/products/hornet works for a Golf sized car or the Talon) for the car based parts.. so you don't have to break the bike down again and again... and find a decent bike shop on the last day or so and repack.
I think you might be better off this way as opposed to renting a larger car to carry the hard case and bike. Larger cars tend to be disproportionately more expensive here than a VW Golf sized machine....
Regardless great to know you want to ride, France can be a magical place for it.