Some more thoughts:

As labelling could be problematic (depending on how much information you wanted to include at each point and the density of objects in the area), it might be advantageous to label each object, whether star, blackhole, etc. etc., with a two digit identifier. Then around the outside of the overall map, you have boxes (frames) for each object. Within that frame, you could include whatever other pertinent information is appropriate, i.e. star type, political allegiance, population, number of planets, and whatever else you are interested in.

For multi-planet systems, you could even do a 2D view of each solar system showing the number of planets, their distance from the star, etc. within their frames.

Back to the whole general concept of my representation: One problem that I see with it is that if there are incredibly large objects which cover a large swath of the area (rather than objects that are basically points such as stars, etc), it would be difficult to represent their position/shape within the spherical 3D space. Although you could just pick some points along the length of such an object to closely approximate it size/shape.