Yes you can use an Apple TV connected to the HDMI cable. That HDMI cable only goes to the main TV above the door.
The Apple TV will be pretty useless, though, if you attempt to connect it to campground WiFi. You'll have better luck using a local hotspot or shared WiFi from a phone or tablet (but be warned you'll consume a bunch of bandwidth in that case).
If its the new Apple TV you'd still be able to play many of the games (at least the ones that don't require internet access).
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