2nd in command to sauron

[Eurytus]

The Nazgul would be in line before some man-turned-sorceror. The WItch-King was in charge of Minas Morgul; Khamul was in charge of Dol Guldur. The Mouth of Sauron was simply Sauron's Lt. at Barad-dur, and I can't see him leaving and being given his own fortress to reign over the much more active and powerful Nazgul. It doesn't make sense.

I disagree. Khamul resided in Dol Guldur yes, but he did not lead it's armies or do anything of a commanding nature.
Tolkien states that, of the Nazgul, only the Witchking had the ability to function using his own iniative without the will of Sauron behind him. Something that would be critical for any commander.
This is proven by the fact that when the Witchking died the command of the army of the Pelenor fields passed to Gothmog. Whom we know little about but since Tolkien only states 2 Nazgul names (the Witchking and Khamul) it is reasonable to assume he is something else.
Once that battle was over, at the parley it is stated that the Mouth of Sauron is now Sauron's lieutenant. I see no reason to doubt that. The remaining Nazgul do not have enough initiative to be able to command themselves.