Hermaeus Mora is the Demon of Knowledge, Lord of Secrets, whose dominion is over forbidden knowledge. It stands to reason, then, that he would know the most forbidden of all secrets: The existence of the Godhead and CHIM. Now, as a Daedric Lord, he is obviously self-centered enough to avoid zero-summing upon discovery of the knowledge. Given the power of CHIM, he would probably use it.
Further evidence of him knowing CHIM is found in the Black Books. Waking Dreams of a Starless Sky is chock full of dangerous knowledge about the Godhead, and seems to talk about the Observer portion of the Enantiomorph in particular("The eyes, once bleached by falling stars of utmost revelation...). It's a Black Book, so we can safely say he knows its contents. Knowing of the Godhead and the illusion of reality should make someone zero-sum, but Mora obviously didn't so he probably has CHIM.
I think it goes further than that though. Maybe, just maybe, he knows about the Amaranth. Think about it. The road to becoming Amaranth is "Take part in enantiomorphic event as the Observer->Become sensory deprived from the maiming->Begin to dream your own reality", if I understand it right and we take the Children's Anuad as the source.
Now examine Hermaeus Mora's actions in the Dragonborn DLC's main questline. It's a textbook enantiomorphic event, notably different from the Civil War enantiomorph in a number of ways.
1) The LDB is the King, not the Observer.
2)The final battle takes place on a tower.
3)The imagery of the Champion taking the Rebel's heart out is present. Notably, Hermaeus Mora is the Champion.
Think about it. Mora had absolutely no reason to kill Miraak there. He could have let you lop Miraak's head off right then and there. In fact, there really isn't a reason he needed the LDB in the first place. He could have simply zapped Miraak on his own. That doesn't happen though. Instead he steals your kill and stabs Miraak through the heart almost exactly like the famous enantiomorph of Auriel, Trinimac, and Lorkhan.
It goes further than that, though.
We know that Hermaeus Mora "scrys the tides of fate" and time is very malleable for him. What if he set the events in Dragonborn up? It should be within his ability. Set up a Rebel in Miraak, a King in Dovahkiin, and make himself the Observer.
Is it possible that Hermaeus Mora is attempting to become a new Amaranth, or some other high-metaphysical power? Or perhaps I'm just crazy...