Where are the most authentic "French experience" hotels in this region, and is it possible to get around on public transportation?
I am not sure how to respond to ¨French experience hotels¨ but, as a general rule, having a car to explore this region would be highly desirable. Trying to always travel by train (bus service is very limited) is very time consuming and can eat up an enormous amount of your valuable vacation time.
LIttle-known rules of public transportation in Europe:
a) Not every town/village has a train station.
b) Just because town A and town B have train stations, there's no guarantee that it's easy to go from A to B. You might have to go all the way to F to transfer and backtrack.
c) Public transportation, outside of big cities, is not there for tourists. It's there for schoolchildren and local commuters. As such, there are bus routes early in the morning, late in the afternoon/early evening, and a few routes at lunchtime. It may not be frequent or even regular -- and on weekends, holidays, or during school breaks, many bus routes don't run at all.
Rural Provence ( e.g. Roussilon) is ,as the above say, not easy to get around by bus or train. The big towns i.e Avignon, Aix en Provence, Arles and Marseilles are well connected - Marseille has metro (subway) service. On the other hand the Cote d'Azur (aka The French Riviera) - Nice, Cannes, Antibes, Monaco has excellent cheap and regular bus and train services.- at least along the coast if not smaller inland villages.
The French are the biggest tourists in their own country so , outside some de luxe hotels in Cannes or Monaco the majority of people you will bump into will be French.
We stayed in Le Clos du Buis in Bonnieux. Very authentic French. It is a large house, transformed into a B&B, run by a wonderful french couple.
I also found the town of Bonnieux to be authentic and not over-run with toursits, and from there we visited Roussillon, Lacost and all the rest. Roussillon is pretty but is very torusisty- if you want an authentic french experience it may not be a good choice
IMO...before you look at accomodations, you need to decide whether or not you are prepared to rent a car. If you are, contine to explore accomodations in the Luberon. If you aren't prepared to drive, look to stay in Avignon or Aix.