1. Go to LinkedIn. Find a mid to high level finance executive at the firm. Add. Explain. They know some of their processes are somewhat shitty and might have the grace to pull some strings / sent some internal cold mails (in ultra large firms, mailing far outside your sphere feels like cold sales) to get things resolved.
2. Sent a snail letter to the CEO. The CEO gets little post that isn’t outright spam. This gets read by an assistant and passed down the chain. Down the chain gives very large urgency. The assistant to the CEO doesn’t know anyone five steps down. So you are coming in like a champ.
Both tips would work in my firm. It’s what I’ve use once or twice to get proper service without lawyering up. The best response was when the premier customer oriented person (director of operations) of a very large firm called me to pretty much complain that the CEO tasked her with resolving my issue. I had to refrain from laughing. (It was a big problem for me but quite a delicate matter for them I explained in kind writing.) She was telling me how happy I should be that she called me. I told here thanks for calling but you should be happy this isn’t in the papers. End of kind call, start of solution.