Welcome to FT
when this is one one booking, you will be picked-up with the limo at the gate of your inbound flight, transferred to the P lounge and then same to go to your connecting flight. When you finish your trip in Paris, you are also picked-up at the gate with the limo and driven to immigration and they wait with you at baggage claim.
In your case, I would very strongly recommend you to call la Première line before your trip and let them know about your 2 bookings. I am almost sure they will accept to give you the same treatment as if all was in the same booking but you need to let them know before. Ortherwise they have no clue about your “connection”.
Please report here how it went.
Enjoy your trip in P