The rental agencies in the big train stations (Nord, Lyon and Montparnasse) are the most likely to be able to guarantee an automatic.

Choose either the west (Tours, La Rochelle) or the Provence. For the Provence it might be easier to take the TGV to Avignon or Aix and pick up a car there. With early booking, the train might even be cheaper than driving taking into account the cost of the rental, fuel and tolls. For rentals, look into autoeurope.com or kemwel.com, sister companies that are brokers. Their contracts are very clear and offer a good point of reference if looking at other rental options.

You can get to Provence, or La Rochelle, in a couple/few hours now on the train. I certainly would never drive from Paris to either Provence or La Rochelle, but then, I'm not a fan of French autoroutes and love the train system.

Also, La Rochelle and Provence are radically different types of places, so it's hard to comment without knowing how you chose these two areas. I know both of them quite well, but would find it difficult to recommend one or the other without knowing your reasoning for picking these two places out of the myriad offerings of France.

I don't know as much about French spec cars but in the UK, it's become more common for larger cars (which are usually higher spec) to be automatic rather than manual. I would expect car hire agencies to follow this customer expectation.

If the same is true in France, you might expect to have a much easier time getting an automatic if you are OK paying for and driving a mid sized/larger (by European standards) car. I would expect that the smallest and cheapest cars still are largely manual. Maybe this is stating the obvious.

I agree with kerouac that your best luck for an automatic will be at the train stations. Given the heat this summer I would almost wait and see in terms of the weather where to go-if it continues to be hot I would head to La Rochelle and the Atlantic or maybe even Brittany-Vannes is lovely and so is the Dinard area. If it isn't quite as hot then I might consider Provence, although we prefer the area more west of there that is Languedoc-Uzes, Nimes etc. In general that area is much less crowded and cheaper than the heart of Provence with a very similar kind of vibe. If you took the train to Avignon TGV station you could pick up a car there and explore a bit of both areas.

You might also consider using autoeurope since as a consolidator they should be able to guarantee you an automatic. We don't use them a lot since we lease a car on the Open Europe plan but I think they should be able to guarantee an automatic transmission car. This might be even more important if you decide to take the train out closer to your destination to pick up your car. And with only a week I think you might want to consider picking up the car from somewhere nearer your destination. We are big road trip people and like to drive but the area getting into and out of Paris isn't very scenic and can be a really hassle in terms of traffic.

It's true that you can't get the smallest cars in France with automatic at rental agencies, but you can easily get "compacts" with A/T. Any agency/Autoeurope will show you the listings as to what is available, though, by car size.