I connected the DAC to CD1, and then played my test tracks again. I kept the RCA clear tops in the Eclipse. The volume on the Eclipse was still around 50%. This time there was no harshness or exaggerated treble response. The sound was clean and well balanced. No sibilance at all.
So is it the case that the increased treble that I heard on my first test was entirely due to the DAC over-driving the Eclipse? It certainly seems to be the case, and nothing to do with the RCA clear tops at all.