I need to get a shuttle from ORLY to the Hyatt Recency at CDG. I know there are regular bus shuttles betwee the two airports. But, if I need to get a taxi from CDG to the Hyatt, would it be better to just get a private driver from ORLY to the Hyatt at CDG
Why?
Why are you staying at an airport hotel? When do you arrive at Orly? Are you getting a flight out of Charles-de-Gaulle, if so when and to where?
What are you trying to achieve here?, you are flying (I presume) from XXX to Paris Orly, and the following day flying (I assume) from Paris Charles-de-Gaulle to YYY. Why do you need to stay at Charles-de-Gaulle? Why not fly direct?
Sounds like you are wasting a whole day and night staying at airports, is there not a better plan?
It depends on what you mean by better -- shuttle and taxi will be cheaper. If you take the shuttle, it will cost around 35 euros (21 euros for the shuttle and maybe about 14 euros for the taxi from CDG). Taxi from Orly to CDG should be around 50 euros. Taxi fares are subject to traffic. I wouldn't arrange a private car service. I would just take a taxi if I had to make this trip. Also, you might get more responses if you post this question in the France forum.
Le Bus Direct will take you from Orly to CDG for 18 EUR, departures every 20 to 30 minutes, 70 to 80 minutes travel time.
http://www.parisaeroport.fr/en/passengers/access/paris-charles-de-gaulle/public-transport/le-bus-direct-line-3
From the Terminal 2 TGV station, Hyatt Regency has free shuttle service to the hotel.
https://parischarlesdegaulle.regency.hyatt.com/en/hotel/our-hotel/free-airport-shuttle.html