A quick and easy way to check if your brand name is already registered in France, is to check the INPI online database. Check out the recherche par nom de marque page and simply enter your brand name in the first box, click on rechercher and cross your fingers. The result page will show who has registered the same brand in France or will list similar names with a slightly different spelling. I tried baba and got 81 results with various spelling!
Once you have confirmed that your brand name is available in France, you can register it online with INPI. It will cost you about 250 euros for three categories, e.g. instance magazine, print and cosmetics. Count an extra 40 euros for extra categories. The good news is that the INPI website is in English. You'll need your SIRET number in order to register your brand.