More specifically? When I layed it on as a texture using the Materials tab/ create Custom material and then attach the image to Color as a texture, it distorted where the surface curves.
To get the UV evenly distributed you can do a couple of things -
Just Rebuild the surface to a lot of points. - Rebuilding evenly distributes the UV space. However it makes the surface more complex in this case and inexact if that shape is critical, so…
Rebuild a copy of the surface and use the copy as a custom mapping object on the original.
In this case, it looks like you also need to split that surface on the vertical isocurves and rejoin into a polysurface before applying the custom mapping object - that looks like a bug, to me.