Gunwolf,
You seem to have described hunting an Auerhahn rather than Birkhan.I'm not sure about Austria, but in Germany Auerhahn were usually hunted like you described(when they could still be hunted) and Birkhahn were usually hunted on the ground, during their bodenbalz.If close enough they are shot with a shotgun;with a rifle only if they can't be called closer( hence the blackcock on a shotgun).If its different in Austria, disregard my comment.
Mike