"You might consider informing your client contact of the dysfunctional payments department, but I haven't seen that produce reliable results (again: the incentives of the procurements or payables people aren't aligned with those of your client contact)."

This is good advice. I'm a PM in a large firm that engages vendors/suppliers. If a supplier that was important to the project was having payment issues I'd certainly escalate to a contact in Payments. It might have no effect :-) but I'd have a go.