I made that trip last year and it's a good one. I think you'd be all right with dates after Christmas.You'd arrive in Santiago the 27th and I don't think the dates would affect that schedule: city tour, optional to Valparaiso. I can't imagine restaurants being closed down at that time or shops. There is a good shopping area Plaza Bellevista, a short walk from the hotel.
You'd go to Puerto Varas from there and stay at the Cabanos. We toured a fish market and visited a museum, which might be closed but I think GCT would find something else. If restaurants in the town are closed, the restaurant at the hotel is good and offers a great view of the sun setting over the lake and the volcano.
The only day that things would be closed would be New Year's and I think you'd be in Bariloche at the Cacique. It's an interesting area. I'm pretty sure there was a free day there so they may just make an adjustment. It's a nice hotel, good restaurants, a swiiming pool so I thinIk you'd find something to do.
You'd be in Buenos Aires after Jan 1 so that schedule would likely be unaffected. The Lancaster is about one block from Galeries Pacifico (shops and food court) and Calle Florida, with shops galore -- can't imagine that being closed.
The only question I would have is about the PD but they may have some sinlges who are willing to work over New Year's.