You can project any shape with no distortion if that shape could be printed on a ( 2D ) piece of paper and then folded smoothly into the desired 3D shape. If you can fold it into the 3D shape there exists a projection that can leave no distortion. If you cant without stretching or cutting the paper then you will always have distortion and its more of a choice of what kind of distortion do you prefer or where do you want it to dominate.

For a perfect cone you could map it as a sector of a disk where the ends of the segment wrap back on themselves. If the mountain is not perfectly conical then it cannot be made into a perfect sector of a disk but I would think a map based on a sector of disk is a good starting point.

You could make something like a Mercator projection for a cone by using cylindrical coords. The top of the map could be at the center of the cone and at the highest elevation. Then the bottom of the map would be the base of the cone. The map would be stretched so that the higher elevations would have much larger map area than the lower but it would fit on a rectangular piece of paper.

The first is almost equal area but not quite when the mountain is far from a perfect cone. Also, when the code is not perfect then neither of these two would produce a 1:1 mapping between mountain and map tho this is true for all maps of irregular objects.