The tunnel takes the exit; the proxy keeps the address

Your scraper has to leave through Mullvad. In the container you install wireguard-tools, drop the provider's config in /etc/wireguard/, and run wg-quick up.

Egress moves: the exit is the relay's address; the server's own IP stays hidden. Then you reload the terminal URL, the one that should have died when 0.0.0.0/0 went into the tunnel. It answers in 33 ms. Hoody keeps replies to inbound proxy traffic on the path they arrived on, so a full tunnel and your published URLs coexist. Three down/up cycles later, still 200.

What breaks this is a VPN app's kill switch: a default-deny firewall that drops the request before any routing decision exists. Run the config file, not the app. Add an SSH key first anyway; SSH is brokered by the host runtime and never crosses the container's network.