Camping Les Peupliers, at Riviere sur Tarn - not quite where you asked for, but a fantastic area. The site is good, nice big pitches, slightly shady in some parts, the toilet blocks are good. There is a small shop on site, and a bar and restaurant, and you're within a short walk of a Boulangerie and Butchers, and the village of Riviere sur Tarn. The Gorges du Tarn are on the doorstep, and there are absolutely loads of interesting places to visit. The site is alongside the Tarn river, and there are beaver in the river which can be seen at dusk. Overhead you may see vultures flying, and all around there are lovely wild flowers, birds and butterflies. There are some super walks from the site - nothing too strenuous - and then some more energetic walks up into the hills. The site has a smallish pool with sun terrace.
We may be back there ourselves at around the same time. We've been quite a few times, but it's a site and an area that we never tire of.
Camping Indigo at Royat in the Auvergne is another site, which is within (steep) walking distance of Royat - and again a lovely area, near to Clermont Ferrand and the Puy de Dome.
Both areas are around the same 'distance south' as Lyon, but perhaps not in the places you really wanted to be. I can't say we've stayed around the Lyon area except for overnight stops, briefly, in the past. I think the site at Tournon was recommended by someone we met, but we've never stayed there.