Tank you for the comment! I´ve added some information on how to get there (also see the two links I suggest at the bottom of the mounatin page). Unfortunately, I couldn´t find out about the hut, not even on the Mittenwald homepage, which lists several huts in the area. Sorry!
"Got tight last night on absinthe and did knife tricks. Great success shooting the knife into the piano. The woodworms are so bad and eat hell out of all furniture that you can always claim the woodworms did it."