This isn't canon, but this is how I've always imagined it. All that light on the bridge has to come from somewhere.
Hmm...it may not be canon in the sense that we did see it, but also it's not necessarily a contradiction, because the shot with the noticable ceiling is from the second pilot film and therefore before
the Bridge underwent several upgrades and changes.
Thus, for the regular series, the inner Bridge dome may have looked like what you designed, we just never got to see it.