There are places where you can do this on New Year's Eve for $60 per person---but the transprtation costs between Paris and the restaurants in Back of beyond-istan where you'll find that sort of deal will probably be prohibitive.
Pick up a couple of bottles of good champagne at a local wine shop in your travels, stop at a neighborhood Monoprix and buy a couple of platic flutes. Then put them in a tote bag with some cheese, fruit and a fresh baguette. Take that bag to any good little neighborhood restaurant where you can get a table for dinner on New Year's Eve. Then after dinner & dessert, take your tote, filled with New Year's treats, down to the Pont Neuf or Pont des Arts (or just about any pont across the Seine) and join the groups there awaiting the twinkle lights and fireworks at the Eiffel Tower at midnight. No doubt,you'll become a part of someone's party---or they'll become part of yours---within minutes of popping your first cork. Be sure to save some champagne for a late-night stroll along the Lefy Bank quais along the Seine (hand-holding mandatory, kissing under every bridge highly recommended, singing & dancing optional.