With those programs... you can not.
Turning a flat projection into another flat projection, intended to be folded into a 3D solid requires some transformations that are not easily achievable.
There is a chance Hugin can make some of those transformations. I think it can turn that spherical projection into a cubical one, but I am not sure if you can choose another solid.
One option could be using Blender which is free.
Project the texture into a UV sphere.
Cut the seams.
Edit and peal methodically the vertexes into a flat arrangement.
But as you can see it is not a "non" professional work.
You can also try to model a 3D object and let this program to peal it for you https://tamasoft.co.jp/pepakura-en/
Some other options here: https://alternativeto.net/software/pepakura-designer/
P.S. Gimp and Paint net, or any program for that matter can be professionally used.