Hi again!
Thanks a lot for your comments, Stefanie and Bernd. Yes, I also took these postcard-pictures for research, when I looked for the Sanatorium. I really wonder, why nobody from the community of Kochel knows about the place, it should at least be in the land register, or not? But when I asked, I got the same answer two years ago.
Walchensee is part of the community of Kochel, but from Kochel you still have to drive twenty minutes on a mountain serpentine road to reach it. Walchensee itself is a tiny tiny place (250 inhabitants) and really seems to be the end of the world. An unbelievable beautiful end, however.
As C. Callinicos mentioned in "The Mario Lanza Story", they stayed in cottages. So it did not seem that far-fetched to me that the house is the house. But of course, how does one know that?
Halfway between Kochel and Walchensee is a small spot, called Urfeld. At the shore of the lake there is a huge hotel, nowadays an abandoned place, that also claims to have hosted Mario Lanza.
https://www.sueddeutsche.de/muenchen/wolfratshausen/geschichte-eines-hotels-wo-kirk-douglas-und-mario-lanza-logierten-1.2946476(I am sorry, but this article is in german. It mentions that Mario Lanza visited this location during his stay in Walchensee from time to time to eat his fill. Well, I guess there are lots of stories like that).
Bernd, I add a picture of the map of Walchensee. The house is one of the last on the road "Am Tanneneck".
And yes, on my journey I took my old travel gramophone with me (which was "born" in the early 50ies and still is in brilliant condition) and a collection of Lanza-records. Sure there are more practical methods of sound reproduction, but this one led to lots of interesting encounters and opened many doors during the travel.
Good luck for your journey to Walchensee, Bernd, and it would be nice to read about your experiences!
Greetings from Berlin!