Go to the open markets for shopping and to buy food for a picnic.
Open markets in Paris: Paris Flea Markets and tips
Outdoor Markets and Hours
Hang out at the Champs-Elysees or Montmartre for New Year’s Eve.
I’ve never been able to talk my husband into one of those little cruises on the Seine, but that might be a festive activity. Of course, you have to dress warmly for all of this.

Your destinations are inconsistent with a budget tour, as all are among the more expensive, although certainly pleasant, destinations. And six major destinations in 15 days is pushing it.

In Lugano, we found the Hotel Zurigo a pleasant surprise in terms of cost and size of rooms, but they have no restaurant, and breakfast was, if I recall correctly, coffee, juice, and a roll, but that’s probably healthier than a bigger breakfast.

We’ve been to Paris, Rome, and Luzern, but I know of no budget accomodations there (we often stay in apartments for a week or two, which can save money). In Luzern we stayed at Hotel des Alpes in a room with a balcony overlooking the river and the chapel bridge. They had a nice bigger breakfast in a room overlooking the river, and while they were not expensive, they were not budget either. In Zurich we stayed at Hotel St. Josef, which was near old town and very reasonable, with breakfast, but Zurich itself is quite expensive (but worth it if you like art). Many of the things we liked, such a boat trips, may not be available in winter, but the Christmas festivals should certainly be entertaining.

I post trip reports in the forum area of Fodors.com under the name clevelandbrown if you need more information. I include the prices we paid, but there are more than a few reports and they tend to be lengthy. If you are interested, look them up.

I would seriously consider shortening your itinerary. Lugano is lovely, but probably better in the warmer seasons. Rome and Paris are memorable, but I think it takes more than two or three days to appreciate them. A lot of people don’t appreciate Zurich as there are no mountains there. Everyone loves Luzern, and few have even heard of Lugano (the Italian lake district is popular, and I would consider Lugano part of that).

You could free up a day by flying into one destination (Paris, perhaps) and flying home from another (Zurich or Rome, perhaps). Search for multiple destination flights, rather than round trips, to find these.

If budget is a really big concern I’d skip Switzerland; it was the priciest place we went this past summer on a trip that included Paris, London, and Florence. Of course, you may just want tips on the cheapest way to do the itinerary you’re planning. It does seem a little rushed to me.

After living in seemingly THE MOST EXPENSIVE PLACE ON EARTH (Norway) for 3 years, I found Switzerland every bit—if not more—expensive. They can be done on a “budget” but that requires very good planning. Lodging, meals, and transport can be very expensive, so do your homework on those and plan well. In addition to the most recent editions of guidebooks (check library and bookstore), also check Tripadvisor. For hotels, consult Tripadvisor then try your hand at Priceline. Biddingfortravel.com might give some more specific bidding guidance for hotels in your specific areas.

I hope that $2000 is after you paid airfare and hotels! Maybe cut down on some of the cities as other have suggested. Plus, try to swap out expensive cities like Zurich for one equally as nice but much cheaper like Munich.

I wouldn’t say skip Switzerland, if the OP has he heart set on that country, but do expect higher prices. And is it absolutely necessary to go over Christmas? Christmas-New Year and summer are the two most expensive seasons for flights to (and within) Europe…maybe consider a different time of year to get cheaper airfare?

Yes.
I was halfheartedly looking at Christmas in Paris and it was far cheaper to go just a week before or after Christmas.
But I understand that sometimes the whole point is to get away during the actual holiday.