Audio MIDI multi-output device: Airplay greyed out/showing 'Offline Device'

This is similar to the question posted in 2022: audio midi multi output device : airplay … - Apple Community

I am attempting to create a multi-output device through Audio MIDI setup on my MacBook Pro (2019 16" Intel, Ventura 13.5). I would like to output audio to my headphones connected via 3.5 mm headphone jack, and my Apple TV 4K (Model A2169) which is connected to my LG TV via HDMI. The connected Airplay device keeps becoming greyed out and disconnected as I try to play music through it. Steps are outlined in more detail below:

I am able to add my headphones to the multi-output device without issue.

I can also click the + Icon to Connect Airplay device, and I see my Apple TV 4K (Living Room)

The connected Airplay device is shown in my list of audio devices (sidebar) and also as a selectable option in the multi-output device:

However, when I choose my multi-device output as my output in my Sound settings:

And I attempt to play a song through Apple Music, the Airplay device becomes greyed out in Audio MIDI Setup, the Airplay Audio device disappears from my list of Audio Devices (sidebar) and no audio is sent through it:

I am able to re-add the same Airplay device again (Apple TV 4K - Living Room) with the + icon, and the original greyed out Airplay Device changes to: "Offline Device"

Attempting to change the sound output to the Multi-Output Device again results in the same issue, where the Airplay device becomes greyed out and no sound is produced:

Any tips on how to troubleshoot this further or what the issue could be?