Which is correct regarding the supplication, "O Allah, judge with truth," which the husband repeats, and is it better to replace it with, "O Allah, overlook our faults and forgive us, for You are indeed the Forgiving, the Merciful"?

If, by his supplication (du'a), the husband intends to ask Allah to decree the manifestation and victory of truth, then it is a good supplication, and there is no harm in it. Allah Himself commanded His Prophet, peace and blessings be upon him, to do so. However, if he intends by it to ask Allah to judge him with His justice, relying on his own deeds, then this is a grave error. This is because servants are in constant need of Allah's mercy, and righteous deeds are a bounty and facilitation from Allah, not a price for Paradise. If the husband intends this latter meaning, then your blaming him for his supplication is correct.