Bearded Parrotbill is rather east of Vienna like lake Neusiedler See.
In Tirol I would look for alpine birds.
Golden Eagle, Black Groose, Alpine Accenture, Ptarmigan, Snow Finch, White-backed Woodpecker, Three-toed Woodpecker, Ring Ouzel, Nutcracker, Crag Martin, Alpine Chough.
Good idea is...