Is the husband obliged to pay the full agreed-upon dower (Mahr) and deferred dower (Mu'akhkhar) to cover up the wife's reputation, considering that the marriage took place under duress due to their joint commission of adultery, and that there are other reasons for divorce such as differing madhhabs (schools of thought)?

Sincere repentance from adultery is the most important thing to do, for Allah the Almighty says: (Except for those who repent, believe and do righteous deeds. For them Allah will change their sins into good deeds. And ever is Allah Forgiving and Merciful). The prompt and deferred dowry is obligatory upon you, and the claim of agreeing to it for the sake of concealment does not negate it, because you agreed to it. Allah the Almighty has commanded that women be given their dowries, for He says: (And give the women [in marriage] their dowries as a gift). The verses indicate the obligation to pay the dowry and the prohibition of taking anything from it without the wife's consent. As for her being Shiite, you knew that when you married her, and Allah has commanded justice with anyone, for He says: (And let not the hatred of a people prevent you from being just. Be just; that is nearer to righteousness).