My suggestion for a fall-back name is simply to call the structure "The Cliff." It has historic precedence.
While researching the history of the area, I kept coming across that name in old newspaper articles about the Ocean Beach-Sutro's-Cliff House area. Seems our predecessors considered anything from Ocean Beach to Mile Rocks Beach as "The Cliff," and old newspapers routinely ran stories like 'Nice weather draws big crowds to The Cliff' or 'Trains to the Cliff delayed by rock slide' or 'Balloon Ascension Sunday at The Cliff.'
At first I thought these were references to the Cliff House itself, but it eventually dawned on me that the restaurant's presence gave rise to a shorthand reference to the entire geographic area, what today we'd generically refer to as Lands End.