There are certainly places in France that see few American tourists, but they have often been well and truly discovered by the the British.
I don't rent cars in Europe, so I only visit places accessible by train or bus. There are surely smaller/quieter places than I've seen, but I think you'll need to get away from the coast, and Languedoc-Roussillon (closer to Spain) may work better for you than Provence-Cote d'Azur (closer to Italy). The former area seemed less touristy in general. However, people (including artists) have been traveling to southern France for ages, so you won't be breaking new ground.
Inland from Nice, I found the large town of Vence devoid of visible tourists when I visited it during a June mid-afternoon. It's larger than what you're looking for, but there might be something in that area. Be sure to research the terrain, though. It's very hilly, so I don't know that bikes would be a great mode of transportation. In 2017 a local recommended the town of Tourrettes-sur-Loup to me, saying that it sees few tourists. Unfortunately, I didn't have time to check it out. It's still larger than what you're describing, however. And it seems to be set on a hill.
A bit farther west, I visited the inland towns of Biot (which has a small glass museum) and Vallauris (where Picasso worked on ceramics), and that area was much less overrun by tourists than the coastal towns. But there were still tourist shops (and a Leger Museum in Biot), so you'd have to look at places a bit more off the beaten path. One advantage of this area is that you'd be able to drive a reasonable distance from your rental and pick up public transportation to Nice (bus or bus/train combo) if you wanted some big-city activity and didn't care to drive and deal with parking issues.
Still farther west, I didn't hear English in the town of Hyeres, but it's very near the coast and sort of a jumping-off point for the Isle of Porquerolles, so there are a lot of people passing though. I'd guess most of them are French. But you might find something inland from Hyeres.
But do some reading on Languedoc-Roussillon, where I think you'll have more options that fit your preferences, and I suspect at a lower price point. See if you can find a copy of the DK Eyewitness Guide; it will have a little write-up on a lot of the smaller towns, and I've found it good about listing market days, though those should always be verified online. Perhaps a village in the Pyrenees, but not at altitude so you can reasonably count on warm weather? The Yellow Train from the (hyper-touristy) town of Villefranche-de-Conflent to Latour de Carol is a very scenic ride, but it draws tourists, so you'd want to pick a base not near one of the train stations.
I've spent a good bit of time on the other (western) side of France and enjoyed the picturesque towns, but the weather tends to be a bit more unsettled there.