They also had many, many, other strategies to continue confusing people.
For example, the carriages actually had adjustable covers from within, so they could change the logos around them every few streets.
There were many other carriages going around in the City back then, so it wouldn't be easy to trace their routes. Even if they did, who would say they were the ones at the party? It was a City; the number of beast carts was no joke.
That said, all carriages that exited during this time would've probably been monitored. This was one of the sources of aether letters, after all, so it was expected for them to have better communications than others who'd probably take a few hours to recover and regain bearings after what they had done.
It was why they rushed so much to get out, even risking bumping into other carriages, which ultimately added them to the suspicious carriage category. However, it was still better than being locked in.
