Is working as a financial auditor in a government agency that collects fees and deposits them in usurious banks for interest—with the accountant recording this interest and the auditor reviewing it—lawful or unlawful?

Your work in auditing both usurious and non-usurious accounts in a fee-collection authority is not free from aiding in sin. It has been stated in the Hadith: "Sin is what wavers in the soul and causes hesitation in the chest, even if people give you a fatwa and give you a fatwa." Therefore, you must leave what makes you doubt for what does not make you doubt, and seek permissible work. For whoever abandons something for the sake of Allah, Allah will compensate him with something better. And "Whoever fears Allah – He will make for him a way out And will provide for him from where he does not expect."