Yes, I understand the double work stuff only too well, Sascha.
A one way step I’m aware of, with the options to adjust the “distortion” of the projection accordingly is presented in the clip below, please find in the same link a scene file of it.
The dome could have less segments, which allows to have some of the “rings” scaled and use a SSD 9SubSurfaceDevision) to get the needed precision in the mesh. I prefer the simpler version as below. Let me know if there is something you miss. As I wrote in the other thread, the UV data is the key here.
(Perhaps you get the script mentioned, or find/create an action script in Photoshop to do that [Polar Coordinates]).
About the LW idea, use it if that works better for you, of course.
If you like to voice your need directly, please use the Suggestion form, that is the direct link: (Here the German version Vorschläge)
The clip and the scene file:
All the best