Your idea of going via Bordeaux seems to me the most convenient from a scheduling perspective.
From Bordeaux, you will need to take the train (to Montparnasse station in Paris) and then go to Orly by taxi or public transport.
Otherwise the Bus line 335, which runs from Sarlat to Souillac, is extended every summer (July/August) and the bus continues to Brive-la-Gaillarde Airport.
The timetable is below:
https://www.aeroport-brive-vallee-dordogne.fr/sites/www.aeroport-brive-vallee-dordogne.fr/files/FH%20LIGNE%20335%20WEB%20DU%2006072024%20AU%2001092024.pdf
The column headings indicate the days of the week the bus operates.
L = Monday, M = Tuesday, m = Wednesday, etc.
A bus ticket costs €2.50. A taxi will cost 50 times more (approximately €120)
Note that the bus schedule is synchronized with the train schedules departing from Souillac, particularly those going to Paris (Gare d'Austerlitz).
Anyway, check if the bus schedule matches your flight schedule.
There are other options involving taking a bus, then a train, then a taxi, but since you're looking for the simplest solution, I won't mention them.