• I think four points is sufficient to find the location of the peak (but I've failed to figure out how in my tired state), and
• I'm pretty sure a fifth point is necessary to find the height of the peak (since we don't know the radius of the mountain).

Of course, the above is rather useless without being able to back it up. The purpose of this post is really to provide the drawing I've been using (in part) while trying to figure out the solution. ABCD is the grid you presented, and P is the peak the mountain.

Update: Struck out what I realize I really do not know. There is definitely at least one case where the four points is insufficient to determine the height (when they are all at the same height), but that's all I know.

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