Render mode for half-dome projection, given a certain UV unwrap

I am making an exercise of rendering a simple 3D scene and placing rendered textures onto a 3D model of a dome, that looks like this:

I’ve rendered a UV unwrap of the dome and manually dialled in sizes of each square piece so that they fit the unwrap, which helped me with deciding on the aspect ratio each render should have.

Now I finally have all the textures wrapped like so:

But I am not sure about the half-dome part of the model, as the txt is warped and the lines as well:

I’ve rendered the scene in a 2D mode, but maybe I should have used fish eye mode for that part? Is it possible to understand which mode to use on the spherical part from the UV unwrap of the model? (I did not create the model and its uv map) Thanks for help!