Extend, not mirror. Plug the iPad into the TV with the USB-C to HDMI cable, turn Stage Manager on, and set the external display to Extend. Mirroring would put the host console — every answer on it — straight onto the TV.
Host console on the iPad. Tap Open host console and leave that window on the iPad screen. It is the private one: every answer, the strikes, the timer, the undo.
TV board on the big screen. Tap Open TV board, then drag that window onto the external display.
Tap the TV splash once. Safari only allows audio after a tap, and it counts per window. Skip this and the whole night is silent.
Fullscreen the TV board and set the TV as the audio output. All sound comes out of that window; the host console is deliberately mute.
Check the two windows can see each other. The link pill at the top of the host console turns green. If it does not, make sure both are normal — not Private — windows in the same browser on the same device.
You are now independent of wifi. Nothing is fetched after this point. Put the device on power, set Auto-Lock to Never and turn Do Not Disturb on.