PARIS 5/29 & BASQUE COUNTRY 6/10

I'd suggest renting a car (maybe at Orly) and driving south through the Loire and Dordogne valleys. Amboise and Sarlat would be good bases. Beautiful scenery, Renaissance chateaux and medieval castles, cave paintings, wonderful food. You wouldn't have enough time to thoroughly tour either region but you'd see enough to decide whether you want to return. Drop the car at Bayonne.

There's a direct train from Paris Austerlitz station to Amboise (St-Pierre) that takes 2h. I stayed there once and had the best FR onion soup at Anne de Bretagne. The broth was the best I ever tasted. The sandwiches at Café des Arts are good too. You can take a direct bus to Chenonceaux from Ambiose that takes some planning since you need to know what street to catch it and what side of the street to stand on.
From Amboise you can take a train to Bordeaux St-Jean (2h 30m) that involves one connection in St-Pierre-des-Corps. At least you can take a direct train to Bayonne from Bordeaux-St-Jean (1h 45m).

You could also take the TGV to Tours (St Pierre des Corps) and rent your car from there.

I very much like the idea of a few days in the Loire Valley as long as you like gardens and chateaux. You'll never do them all but one or two a day can be so very restful and pleasant.

I also like Kim's thought of TGV to St Pierre des Corps (the TGV stop near Tours) and then a car for the duration of your Loire Valley visit. It will make things so much easier.

One of my fav parts of France.

Then drop off the car and train to the next Rick Steves adventure...

Since you're heading southwest, I would suggest a base further west instead of Amboise (which, in my very limited experience, is congested to the point of being not very enjoyable -- of course YMMV). If you took the train to Tours you could rent your car from there and have an opportunity to explore the western end of the Loire region. For example, you could spend your first night or two in Azay-le-Rideau (Hotel Biencourt highly recommended) and visit the Fortress of Chinon and the Chateau of Villandry, in addition to Azay's own chateau. Further west, in Saumur there's Fontevraud Abbey and the National Riding School / le Cadre Noir.

https://www.ifce.fr/en/cadre-noir/

This TripAdvisor map shows several castle hotels in the region where you'll be.

https://www.tripadvisor.com/Hotels-g187070-c3-zff16-France-Hotels.html#MAPVIEW