I just re-read your post....
So for clarification, you want to put a spiral on a simple cone--as few faces as possible, no extra divisions--and avoid the warping along the sides?
Normally I'd recommend to you exactly what you already said you tried, to lay the map down in stripes and rotate the points. BUT... if that's too many polys, there is an alternative, but it's a bit tricky.
What you need to do is "bake" the conic projection into your map. You can use POV-Ray to do this.
In a nutshell, with the number of vertices and the type of map you want, there is *no* simple projection that will not have warping. What you really want is a conic projection, but it needs more points to hold up. So, if you can't get rid of the warping in the UVs, the way you fix it is to introduce the opposite warping to the texture. That way, when you apply it, the two distortions cancel each other out and you get exactly the map you wanted.
Here's an example of a sky cube with a spherical warp, so you can see what I mean. This distortion lets me apply what was previously an ordinary cube to a spherical environment map:
A better view, as a checkerboard so you can see what's going on:
To do this in POV, you need to either set up a camera with the projection you want--this is easy for spherical projections, I've never tried a conic projection but I am certain it's much harder to do--or create two versions of your object: a high-resolution version with exactly the UVs you wish you had, and a second object that is geometry built out of the UVs of the first. You then render from the first to the second to transfer the texture to the new UV map.
Since I know what I just said was probably impossible to follow
, here's a picture:
You can use the UV to DXF plug-in I put up on my blog the other day to extract the texture coordinates to geometry:
http://independentdeveloper.com/arch..._uv_map_to_dxf