With the caution that I haven't been to that part of France in late May or early June (my first thought is always about potential weather issues), here are places I found interesting in 2019:
The French Basque Country: I liked Bayonne and St-Jean-de-Luz better than Biarritz but confess it was raining cats and dogs on the day I visited the latter. I didn't have time for any of the mountain villages, but Rick has some thoughts on them. I think they are covered in the book on Spain rather than in the guide to France.
The Spanish Basque Country: More interesting than the French, I think. Lots of options there, but I'm assuming you want to stay on the French side of the border since your time is somewhat limited. If not, ask and we will give you some ideas.
Dordogne/Lot: Most people travel here with a car and see a lot of the lovely small villages. I was stuck with bus/train transportation so spent more of my time in several of the good-sized towns. Allow me to mention that they are worth time, too. Perigueux, Cahors and Figeac were my favorites. I also liked Bergerac and Brive-la-Gaillarde. My visit to Rocambour was very disappointing. Transportation (bus followed by a long walk along a shoulderless highway) took longer than expected so I didn't have time to see the upper part of town where the religious buildings are located. The lower level is nothing but tourist shops and cafes. Historic the buildings may be, but if you don't allow time to see the upper level, I suggest skipping Rocambour.
Bordeaux is a handsome city with a nice historic center. Not the most distinctive architecture in the country. There are some worthwhile museums. The Centre du Vin gets very mixed reviews.
St-Emilion is a wine town often treated as a side-trip from Bordeaux. I dislike wine and prefer less touristy towns, but I quite liked St-Emilion. The vast majority of the tourists stayed down in the lower part of town. I enjoyed the untrafficked streets elsewhere. Do be careful if it's raining the day of your visit. There are cobblestones (slippery when wet!) and steep streets--a recipe for a bad fall.
Moving northward and back to the east, I can recommend Limoges, a city known for decorative arts (museums hold both porcelain and enamels) that has a very nice historic district. Very nearby is the Oradour-sur-Glane, site of a Nazi atrocity. There's an excellent memorial/museum (worth as much as two hours, I'd say) as well as the remains of the destroyed town.
Farther north is Poitiers, another place with lovely historic architecture. In the case of Poitiers there are also some very early churches. Very nice city not often visited by foreigners.
On the coast roughly west of Poitiers is the popular beach city of La Rochelle, which seems to be popular with domestic tourists. Enjoyable and a good base for the Ile de Re, which (conveniently) is accessible by bridge. I took a city bus so I don't know anything about tolls or private-car limitations on that bridge. Between Bordeaux and La Rochelle is Saintes, picturesque and worth a visit.
North of La Rochelle is Nantes, a large city formerly part of Brittany. Few foreign tourists make their way there but I enjoyed it. It's known for the Machines de l'Iles but I didn't have time for them.
Beyond Nantes there are more nice Breton cities and towns, including Vannes and Quimper. Be aware that Brittany can be quite overcast and rainy even in the summer.
