If you already bought the air ticket from Paris to Biarritz, I recommend taking a room at or near the airport, one with free shuttle service back to the airport. Get there first, stash your gear, and reassess your situation. Figure out if you have enough time to head into Paris on the Metro or TER.
IF AFTER YOU ARRIVE and determine you have enough time, you might consider heading into the center of Paris, then picking up one of those hop on hop off tour buses. This will allow you to see all the highlights of Paris from the comfort of a bus. It takes only an hour or two.
You can use the provided guide and map (on the bus) to determine those sites you would like to return to see. While you CAN hop off the bus to see a place, then catch another bus onward, consider waiting and security times to get into most of the popular destinations, as well as waiting for the next bus. I use these buses to get a good "once over" assessment of any new city. I find it an efficient use of my time.
12 Hours is NOT enough time to visit Paris. But, it is IMHO enough time to SEE the best bits of tourist Paris if from a coach / bus.
I would rely on MSPATH's observations and recommendations above, unless you discovered a better idea. I lived 3.5 hours away by car in adjacent Belgium for two years and would frequently drive into Paris with visiting family and friends. So, I have been there a few dozen times. But MSPATH is THE expert on most all things French.
Also, if your schedule finds you in Paris at dusk or after dark, head for the river Seine tour boats. These buses provide a narrated tour of the sights lining the Seine in downtown Paris. In my experience, most start and end at the river just adjacent to the Eiffel Tower. BEWARE THE TOUTS AND PICKPOCKETS! You can take le Metro or the TER trains to get near to the staring /ending place. After the tour, best done at night IMHO, you can take the metro or another train back to the airport and your hotel. This is helped if your airport hotel has a regular free shuttle service. You can train back to the airport and have your hotel pick you up.
As others have said, this is NOT the best way to see Paris. But all things considered, it may be the easiest and safest. It also has you staying close to your departing flight the next morning. Allow extra time for airport security... If they tell you to come 90 minutes ahead of time ADD AN HOUR. Missing a flight or train is no way to start a Camino...
Lastly, if you HAVE NOT ALREADY BOUGHT the second segment ticket to Biarritz, go with the above recommendations and head from the airport to the Gare Montparnasse area. As others have suggested, reserve a room there. Take the TGV the next morning from Gare Montparnasse. Reserved seats are required. So , I recommend either buying your tickets ahead of time (
www.raileurope.com) or heading over to the station as soon as you check into your hotel room to buy them the night before. This avoids delays and a final rush the next morning. Be sure to allow time for security screening at the train station as well.
Then, after you have ensured how to get out of Paris the next morning, pick up seeing Paris using the hop on / off coach and river boat tours, as feasible.
Enjoy!
I hope this helps.