I haven't been to Bordeaux myself, but it's on my list for a trip to western France, probably in 2019. Why do I say "western France"? Because I spent 89 days in France last year and saw only the eastern part of the country, not even including Paris.
Based on my travel style, you won't be surprised that I want to caution you about taking Rick's 3-week itinerary and adding additional places to it just because you have 4 weeks. His itineraries are extremely fast-paced, seemingly designed for Americans with very limited vacation time. You have a bit more time, so you don't have to rush so much. Rick's car itinerary allows a whopping one day plus maybe a few hours for Provence and three days in Paris. The train itinerary allows just two real days (plus the marginal value of some hours on your arrival day) in Paris. A look at Rick's own guidebook will show you how much you are going to miss.
In the unlikely event that you find a particular spot only deserves as much time as Rick suggests, there is almost always a nearby town (accessibly by bus or train if you don't have a car) that is worth seeing. In Normandy and the eastern edge of Brittany I enjoyed all these places that don't figure in Rick's itineraries: Rouen, Deauville, Cabourg, Dinard, Rennes, Vitre, and Fougeres. I reached them all by train or bus, albeit not at a pace consistent with Rick's itineraries.
Another thing to consider is how tiring it is to change hotels every night or two. I believe there are 12 hotels on that 21-day itinerary. It's one thing when you're on a bus tour that is going to whisk you directly from Hotel 1 to Hotel 2 (and allow you to sleep if you want to). It's a whole different story when you have to navigate yourself in a rental car or must spend not-so-quality time in train and bus stations. That pace would be draining for 10 days. Keeping it up for 4 weeks is not a recipe for a particularly pleasant vacation.
In sum: Before adding additional hotel stops, I urge you to read about the places already on the itinerary and figure out how much time you'll want in each one. Setting aside the question of Paris (how much time there is a key decision, and there's massive room for variation), If I had 28 nights in France, I would stay in no more than 8 or 9 places at the most. It would be fewer if I were visiting Paris for the first time, because that would suck up quite a lot of time.
