Winning in a US court is not the end if the other side's assets sit abroad. Enforcement depends on whether the local courts will recognize a US judgment, and many will not do so readily.
Arbitral awards often travel better, because the New York Convention obliges signatory states to recognize them subject to narrow exceptions. That is a strong argument for an arbitration clause in cross-border contracts.
Either way the question belongs in the drafting conversation. By the time payment has stopped, your leverage is whatever the contract already gave you.