Paris to Dordogne

Bordeaux is about two hours by TGV (express train) from Gare Montparnasse in Paris, and Sarlat, the main town in the Dordogne valley, is about three hours by road from Bordeaux, depending on the route you choose. So you could do that in one day if you don't want to overnight in Bordeaux.

Alternatively, you could take a train from Gare d'Austerlitz to Souillac, just east of Sarlat, taking about five hours, then drive to Sarlat in about half an hour. You might want to base your choice on train fares, rental car availability (better in Bordeaux I expect), or how much driving you want to do.

Sarlat is a good base for the Dordogne, and well worth a visit even if you don't stay there. We stayed in nearby Beynac, at Le Petit Versailles, but I don't know if they're still in business. I can't find their website, the latest Trip Advisor review was from March 2020, and there's a thread on this forum from June 2021 recounting some unsuccessful efforts to contact them. Beynac also has the Hotel du Chateau, well located but I've never stayed there, and Sarlat has multiple choices.

There are no rental car places in Souillac, but there are some in Brive-la-Gaillarde.
This said, Bordeaux is a nice city to spend the evening, and has more car rental options, so it is a very sound choice!

Great plan! Bordeaux is quicker from Paris because it's TGV, while Brive is an Intercité stopper. You definitely want a car for the Dordogne.

I can recommend the hotel we stayed at, the Hotel du Chateau in Beynac. Right on the river, beneath the cliff of the castle, and handy for canoe trips, Sarlat, Domme, Lascaux ....