To inspect HTTPS traffic from an Apple TV you need two things on the device: an HTTP proxy pointing at your Mac and the Charles root certificate in the trusted store. tvOS exposes neither in Settings — both are applied with a configuration profile. ProxyBeam, a free Mac app, builds that profile from your Charles certificate: install it once on the Apple TV, enable SSL Proxying in Charles, and requests show up decrypted.
In Charles on your Mac, choose Help → SSL Proxying → Save Charles Root Certificate… and save it in the .cer format.
Download ProxyBeam (free, macOS 15+), open it and drag the .cer file into the window. The app starts a local server and copies a link like http://192.168.x.x:PORT/proxy.mobileconfig to your clipboard.
Open Settings → General → Privacy & Security, highlight Share Apple TV Analytics, press Play/Pause on the remote, choose Add Profile and enter the link. Details and tips: how to install a configuration profile on Apple TV.
Go to Settings → General → About → Certificate Trust and enable trust for Charles Proxy CA.
In Charles choose Proxy → SSL Proxying Settings…, tick Enable SSL Proxying and add a location with host * and port 443 to decrypt all hosts — or list specific hosts. Keep Charles running on its default port 8888, on the same Wi-Fi as the Apple TV.