Suppose that after joining the treasury department of a large corporation, you find out that it avoids hedging because the cost of hedging comes out of the treasury department's budget

What argument could you make to the CFO to get the firm interested in letting you be the firm's hedging guru?

There is something wrong with the firm if losses on hedges are booked to treasury whereas the gains on the underlying assets that are being hedged are booked somewhere else. You should explain to the CFO that hedging is about avoiding losses that would adversely affect the performance of the firm. Hedging involves investing in derivative securities whose values go down when the underlying assets of the firm go up in value, while the values of the derivative securities rise when the underlying assets of the firm fall in value.
It is appropriate for the costs of hedging to be borne by the treasury department, but these costs should be the personnel costs for those who are involved in the process.