I created a half-sphere from a UV Sphere (+ subd lv3 applied).
U / Project from view (top view).
Fit the island into the UV region (Zen UV addon)
Pin the boundary vertices.
Relax the island (Zen UV)
The cyan line is a 1pt circle drawn with Affinity Designer on a 2048px texture.
I would try selecting the perimeter edge loop and scale it on the Z axis by zero, which should align all the lower edges… As far as the Black line, select all in the UV editor and Scale to Inside the edge of your image…
You should check your image again because it is not fully round the Right and Left, as well as the top, squeeze toward the center edges…
The dimensions are accurate 3.46402 X 3.46402 so it’s not the Half Cubic sphere…
Interestingly @oo_1942 suggesting is as far as i can see the the simplest but does not met the center correctly and also does rotatr this thing arkwardly
But when using the project from view (bounds) method (wrongly named spherical by me ) and pinning the outer border and the center; inverting and simpy unwrapping…
…this might be better… maybe scaling it by 0.998 or something becaus eof the not super exact pixels at the border of the circle… ?
Cool…
However, the last hours I did a manual way.
I downloaded a Dome Calibration image and tried to model the sphere and UV´s after that.
So I projected from view and bounds and than I “TO SPHERE” every Loop on Mesh and UV and Aligned the distances and angles to the calibration Image.
Worked pretty well…
Here is the file with 3 Different versions.
And Now I will look into your solutions 
dome_calibrated.blend (2.6 MB)
Yep thats a good way. That should minimize your overall distortion inside the dome.
All Sides Align perfectly now!
And the Iceworld Image seems pretty round to me…
Thanks for all the Input guys
I think it’s the same result. 
fasf.blend (4.2 MB)
Add…
A pin is also required for the center point (coordinates are 0.5 X 0.5)
I am glad that this grew to such a fruitfull matter
What is also Important, that the Loops of the mesh have the right angle to the center. You can see the required angles in the testimage.
I Adjusted them Manually, because they were off a bit.
At least i tried my best 
Might not perfect yet…
There’s a little difference.
The intervals are not consistent, I don’t know how to sort them. 
fasaa.blend (4.4 MB)
Hi tichy,
you could make yourself a little Geometry Nodes tool to unwrap this:
Ok cool,
this looks nice. As I understand it, this would be a geometrynode setup, that can be used as a tool, so used again and again and again, right?
This is promising and I guess very usefull for others!
Yeah, you can work with it the same way you do with any other operator (e.g. the Unwrap operator in Edit Mode)
