Was the husband obligated to pay the amount his parents requested in one lump sum, or in installments, especially since this was his wife’s opinion, and given that his action led to her poor psychological state?

What you did is absolutely correct, as it embodies filial piety towards your parents, fulfills their desire, and eliminates causes of animosity. What you did was the most appropriate and best course of action. Your wife should assist you in this, and we advise both of you to focus on the positive aspects and rise above disputes, which originate from Satan.