No, the Nginx proxy itself doesn’t need to know its network name; that’s only needed by you, personally, during early setup—right after the initial app create.
The Nginx proxy can just refer to app-name.flycast, like before:
https://community.fly.io/t/run-private-applications-with-flycast/20387/4
https://community.fly.io/t/run-private-applications-with-flycast/20387/7
One of the appeals of Flycast addresses is that they do allow controlled traffic across separate networks.