The Mouth of Sauron

[Nils]

I would think that he was considered above the Nazgûl who were simply doing the dirty work for their Master, looking for Rings and commanding armies.

I don't know about that. In Letters, Tolkien stated that by the end of the Third Age Sauron believed himself to be Morgoth.

This is an important clue because it links the Mouth of Sauron to another character. The character I am referring to is Gothmog. One of the meanings of the name Gothmog is "Voice of Goth" (Morgoth). I believe this reference can be found in The Lost Road and Other Writings.

In other words, it is possible that the Mouth of Sauron = Gothmog. Since Gothmog is second in command to the W-K and by this time the W-K is dead, the Mouth of Sauron would be the head of Sauron's forces.

As to the Mouth of Sauron's identity, there are a couple of possibilities mentioned in Of the Rings of Power and the Third Age.