Just a few thoughts. It may be very difficult to open that door if it's frozen shut. Oftentimes it's buried as sometimes the snow goes all the way to the roof. You might want to have a plan B to head back and catch the last tramcar down. Also, those black stone walls will absorb your body heat, so a solar blanket can make a big difference (in addition to a 0-degree rated sleeping bag). Other than that, sounds fun! Just letting you know so you don't have any surprises.
Even though the picture above shows the hut open in winter, check with the ranger station to make sure it will be open. In the past its been locked for the winter to discourage anyone from relying on it for shelter. Last time I was there it had a broken window, too, which would make it "well ventilated" if a Santa Ana condition developed (I'm sure no one would stay there if a storm was predicted!).