[Son of Númenor]

I thought that 1 of them could be the Mouth of Sauron. Is there anything that could be used to try and prove this theory?

Unfortunately no, but there is enough available to disprove the theory.

The Lieutenant of the Tower of Barad-dûr he was, and his name is remembered in no tale; for he himself had forgotten it, and he said: 'I am the Mouth of Sauron.' But it is told that he was a renegade, who came of the race of those that are named the Black Númenoreans; for they established their dwellings in Middle-earth during the years of Sauron's domination, and they worshipped him, being enamoured of evil knowledge (The Return of the King, "The Black Gate Opens", 871).

Although one (or both) of the wizards may well have served Sauron in some regard after the disappearance of the Blue Wizards into the East, it is pretty obvious based on the passage above that neither was the Mouth of Sauron.