Is working for a company that deals with non-Islamic banks – knowing that my role is limited to auditing and preparing budgets without signing off on loans – considered unlawful?

It is not permissible to work in preparing accounts or usurious loans, because it is considered aiding in sin. The Prophet, peace and blessings be upon him, "cursed the one who consumes usury, the one who feeds it, its two witnesses, and its scribe, and he said: 'They are all equal.'" However, if the work is free from that, and it is in something permissible, there is no harm in working with a company that deals with usurious banks, because dealing with mixed wealth is permissible or disliked.