Now normally we would just have contacted the company and asked them for a data extraction so we could cross-reference the data. But since it wasn't clear who was at fault, and we knew it would take weeks for that extraction, we looked for an internal solution first.
Now there was a subsystem in the machine that worked only in Internet Explorer, with an old authentication scheme, that we could use to see the information we needed, so I, being the only person in the team without formal analysis training but having made my way there from a clerk job, knew exactly what to do.
I fired up the old IE, Excel, wrote in 5 minutes a VBA script that did exactly what you described, click there copy that etc, and 30 minutes later we had our extraction, and resolved the issue completely before the packages were even shipped.
All hail Excel.