I usually rent from AutoEurope when I'm in Europe, including France. AutoEurope acts as a consolidator for a number of car rental agencies, so you can plug in your information and it will show you a list of places and their prices for the time that you need and the places you are going to. The price per day will obviously vary depending on the agency, but also the car. Also keep in mind that if you pick up your car in one place and return it in another, there is usually a drop-off fee. It could be very little and it could be a lot, but just be aware of that.
It will then give you the option of choosing automatic or manual transmission and tell you the cost per day and all the other details that you need. Then you choose the option you like, and you will get more details about that and what it will cost and insurance and all of that. It used to be that it was difficult to get automatic transmissions unless you booked really far ahead, but I'm finding that more and more places in Europe now keep more cars with automatic transmissions available.
I like Auto Europe because they keep control over the process and they can be a big help if something goes wrong. I've been renting from them for decades, and I've never had a problem.
Edit to Add:. With regards to renting in Bayeux or Caen, I would just check both when you're on their website. See what cars are available in both places and decide which makes the most sense.