Yeah, any solution is going to require at least egress rules for its traffic. Tailscale is a bit different since part of what it’s able to do is provide access to your LAN, if desired. Cloudflare just needs two ports, but it’s only providing a tunnel from the host.
Swag has to have its own docker network, and the containers proxies through swag have to be on that network. It can’t be bridge or host. Spaceinvaderone did a good video in setting this up and covers that part very clearly, I think. Maybe I misunderstood, but since you said they’re all on the same network, I assumed it was their original network.