June & September are usually both good months in terms of weather and traffic. In terms of mountain scenery, you'll have lots of snow on the peaks in June, but not much left in September, especially in the Pyrenees. I rode up (with panniers) the Col du Tourmalet and Mont Ventoux in late May and was lucky enough to have great weather each time. I would recommend that you NOT follow the Tour routes when you're out of the mountains, however. Those routes will typically be on major roads which are OK on Tour day when they're closed to motor vehicles (other than those really, really annoying vehicles preceding the tour). You would find lots of traffic on them the other 364 days of the year, however. France has a fantastic network of very good secondary roads with minimal traffic which will result in a much more pleasant riding experience.
It's not really the huge undertaking that you're imagining. It's common to find a hotel or two in small towns in France, and the same goes for restaurants which generally have excellent food at prices much lower than in Paris. Hotelkeepers are usually very welcoming toward cyclists. In June or September you normally don't need room reservations.