The power imbalance between us — a 7-person agency — and them — a multinational publicly traded company — is immense. The dollar amount in question is outside the realm of small claims court, which means that we'd end up in actual court, where we'd lose: not necessarily on the merits, but because of the costs of attempting to litigate against a giant.

It's the implication. The letter is a threat that you take them to court, which will cost them at least five figures, and will tie their attorneys with the case. They don't know how much fighting power you got, just make them believe you're willing. No one wants that fight, even multinational corporations.

But it sounds like from your post that they actually intend to pay you, but they're stuck somehow? Get your lawyer to unstick them.