Unroll Geometry as Texture

I have a 3D model cylindrical vase that I want to unroll as a texture/image so that it can be reapplied to a pure cylinder as a texture (I am trying to save on geometry). The 3D model is comprised of multiple polysurfaces. And some geometry spans over the seam of the cylinder. Shaded mode if fine. Is this something that can be done? I do not want to flowalongsrf from the cylinder to a flat surface. That does not work with the geometry at the seam, even if the geometry if split. I suspect that would also distort my geometry.

Here is an basic example:

191017 Temp.3dm (414.4 KB)