I wouldn't want to get into a car and drive to France after an overnight flight.
ViaMichelin.com offers three possible routes to Sarlat. Have you decided which way you'll go?
I'd suggest spending a little less time on the drive to Sarlat and adding a day or two there, to give yourselves more time in the Dordogne/Lot. There are, however, a lot of interesting places along the way.
I think most nearly everyone who has seen both will say the Dordogne is more engaging than Bordeaux--a nice city with some historic architecture, but not particularly distinctive. If you can't find another way to add some time in the Dordogne, I'd go so far as to suggest skipping Bordeaux.
Zaragoza has an attractive historic center and an impressive Moorish palace. The train station is awkwardly located--about a 15-minute walk from the palace and 30 minutes from the historic area. There seemed to be frequent bus service from the station to the center, but the area around the train station seemed bereft of restaurants as of 2016, so it's not an attractive place to stay. The business-class Eurostars hotel in the train station is comfortable, but the food situation would keep me from staying there again. The hotel restaurant was closed during my July visit, I assume because summers are so miserably hot in Zaragoza that there's little business traffic at that time of year. An extremely limited menu was available from room service. I don't know what the situation would be like in September. Be sure you have an air-conditioned hotel in Zaragoza.
Four nights will be very short in Barcelona (another reason to drop Bordeaux). Four nights is the minimum I ever recommend, and that's for people who have no real interest in Barcelona's art museums. Although there are other cities along your route where you'll find art, I suspect the best is in Barcelona except for the Guggenheim--and you've said you don't care for modern art. I am prepared to be corrected on this point, because I am not an art expert, just someone who goes to a lot of art museums.
There's an interesting privately-run Asian art museum in Biarritz. I haven't been to the Museum of Fine Arts in Bilbao because it was closed at the time of my 2016 visit. Some online research should help you decide whether the museums in Bordeaux, Bayonne, San Sebastian, and Vitoria-Gasteiz are right for you.
I especially recommend the MNAC in Barcelona. In addition to paintings and sculptures, it has two unusual collections. The modernism collection focuses on furniture, jewelry and decorative art. Then there are large medieval frescoes rescued from churches in the Pyrenees. But the city has many other museums, art and otherwise. The Picasso Museum is too popular for its own good; the Miro Museum is generally a much more pleasant place to see art.
Another factor affecting the time you'll want to spend in Barcelona is the presence of a lot of highly popular sights for which you'll need to buy tickets (usually timed tickets) in advance to avoid wasting a lot of time in ticket lines and possibly encountering sell-outs. It's hard to rush through a city's sights when you have to make appointments to see six of them (if they are of interest to you). The problematic six are: La Sagrada Familia, Parc Guell, Casa Mila/La Pedrera, Casa Batllo, the Picasso Museum and the Palau de la Musica Catalana (whose English tours might sell out).
Driving from the Basque Country to Zaragoza appears to be faster than taking the train, and trains are infrequent. For the last leg, from Zaragoza to Barcelona, the train takes less time--as little as 1-1/2 hours vs. ViaMichelin's estimate of 3 hr. 15 min. in a car. There are many trains connecting the two cities because Zaragoza is on the Barcelona-Madrid rail line. I don't know anything about the rental-car drop-off locations.