One possibility might be to try for a home exchange (or house exchange). I know people who do this, though I don't know the names of any agencies. You might be able to find some information through Google or another search engine.

One thing you're up against is that the rental laws in France differentiate between the kind of short-term situation you're looking for and a true housing "lease." If the property is furnished, it is treated like a hotel (which may account for those by-the-week rates you're seeing).

A "normal" housing rental involves a lease, usually for a 3-year period. So you're kind of stuck in the middle - especially since the period you're planning on being in the area includes the height of the tourist season.

There are rental and estate agents in the south of France who cater to the English-speaking visitors, and they would definitely be the place to start.