I totally understand your desire to see Paris. It's an amazing city and you should definitely take advantage of being on the continent! Formule 1 is an inexpensive hotel option, (46 per night for a two person room.) It's in the outskirts of the city, but easily accesible by metro. If you're interested, PM me and I can send you a list of monuments to see. Over the years, I've taken dozens of groups of students there, so I have some experience in seeing as much as possible in a short amount of time.
As for flights, unfortunately París is usually one of the most expensive cities to fly into. Since you quoted a hotel price in dollars, I'm assuming you are coming from the States... I would look into flying into París (CDG) and home from Santiago (SCQ). It may be cheaper.
Right now, flying into Copenhagen (CPH) is super cheap. Easyjet has cheap flights from Copenhagen to París and from Paris to Biarritz. However, I would also recomnend the 90 day advance train ticket purchase.