I think rs4 is a bit of a misnomer, its not part of any of their current games but an additional standalone game. From their website:

“RS4 is not a replacement for any of our games, be it Old School, RuneScape or Dragonwilds. Those games have their own dedicated teams, exciting roadmaps and continued investment. We’ll be designing RS4 to sit alongside them as an additional way to experience RuneScape and the world of Gielinor.”

Here is my docker-compose.yml file with sensitive info scrubbed, its been working for me for a few years now. It sounds like the problem you are having is not with Docker but something in your configuration once the container is running. Feel free to message me if you have questions.

services:
  gluetun:
    container_name: gluetun
    cap_add:
      - NET_ADMIN
    image: qmcgaw/gluetun:v3
    devices:
      - /dev/net/tun:/dev/net/tun
    environment:
      - VPN_SERVICE_PROVIDER=
      - VPN_TYPE=
      - WIREGUARD_PRIVATE_KEY=
      - WIREGUARD_ADDRESSES=
      - SERVER_COUNTRIES=
      - DNS_ADDRESS=
      - HTTP_CONTROL_SERVER_ADDRESS=
      - HTTPPROXY_LISTENING_ADDRESS=
      - TZ=America/New_York
    ports:
      - 3129:3129/tcp # HTTP proxy
      - 8388:8388/tcp # Shadowsocks
      - 8388:8388/udp # Shadowsocks
      - 9047:9047 # Gluten http_control
      - 9046:9046 # qbittorent webui
      - 9696:9696 # Prowlarr
      - 7878:7878 # Radarr
      - 8989:8989 # Sonarr
      - 8686:8686 # Lidarr
    volumes:
      - /etc/localtime:/etc/localtime:ro
    restart: 'unless-stopped'

  qbittorrent:
    image: lscr.io/linuxserver/qbittorrent:5.1.4
    container_name: qbittorrent
    network_mode: "service:gluetun"
    environment:
      - PUID=1000
      - PGID=1000
      - TZ=America/New_York
      - WEBUI_PORT=9046
    volumes:
      - /mnt/drive/volumes/qbittorrent/data:/config
      - /mnt/nas/Downloads:/downloads
    depends_on:
      - gluetun

  prowlarr:
    container_name: prowlarr
    network_mode: "service:gluetun"
    image: ghcr.io/hotio/prowlarr:latest
    volumes:
      - /mnt/drive/volumes/prowlarr/config:/config
      - /etc/localtime:/etc/localtime:ro
    depends_on:
      - gluetun
    restart: 'unless-stopped'

  byparr:
    container_name: byparr
    image: ghcr.io/thephaseless/byparr:latest
    network_mode: "service:gluetun"
    init: true
    depends_on:
      - gluetun
    restart: 'unless-stopped'

  radarr:
    container_name: radarr
    network_mode: "service:gluetun"
    image: ghcr.io/hotio/radarr:latest
    volumes:
      - /mnt/drive/volumes/radarr/config:/config
      - /mnt/movies:/mnt/Movies
      - /mnt/nas/Downloads:/downloads
      - /etc/localtime:/etc/localtime:ro
    depends_on:
      - gluetun

  sonarr:
    container_name: sonarr
    network_mode: "service:gluetun"
    image: ghcr.io/hotio/sonarr:latest
    volumes:
      - /mnt/drive/volumes/sonarr/config:/config
      - /mnt/nas/TV:/mnt/TV
      - /mnt/nas/Downloads:/downloads
      - /etc/localtime:/etc/localtime:ro
    depends_on:
      - gluetun
    restart: 'unless-stopped'

  lidarr:
    container_name: lidarr
    network_mode: "service:gluetun"
    image: ghcr.io/hotio/lidarr:pr-plugins
    volumes:
      - /mnt/drive/volumes/lidarr/config:/config
      - /mnt/nas/Music:/mnt/Music
      - /mnt/nas/Downloads:/downloads
      - /etc/localtime:/etc/localtime:ro
    depends_on:
      - gluetun
    restart: 'unless-stopped'