Can the Apple TV 4K send out Atmos or 7.1 surround to Bluetooth headsets in the aforementioned 7.1 format?

Bluetooth headphones with this claim are virtualized surround, as you have strictly only Left and Right, two ears. But the head-related transfer function is taken into the mix equation. For Apple TV, this processing is done partly on the box, and partly with (compatible) headphones hardware. It will send regular stereo to other bluetooth headphones. Listen with spatial audio for AirPods and Beats - Apple Support

Apple TV doesn’t output DTS HD for any configuration.

If you connect your headphones to an audio receiver, then the audio processing doesn’t depend on Apple TV, and then different options may become available.

Hello jasonfromstrongsville,

We understand that you have some questions about Apple TV audio options with headphones. Per the article you found, Play audio in Dolby Atmos or surround sound on your Apple TV, the following setups can be used to get Dolby Atmos:

"On Apple TV 4K with the latest version of tvOS, you can listen to music, movies, and TV shows in Dolby Atmos. Select one of the following setups to learn how to play sound in Dolby Atmos or surround sound.

• One or two HomePod speakers*
• A television that supports Dolby Atmos audio
• A Dolby Atmos compatible sound bar with a television that supports Dolby Atmos audio
• A Dolby Atmos-enabled AV receiver"

Spatial audio is apart of Dolby Atmos, with the requirements listed here: About spatial audio with Dolby Atmos in Apple Music

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