Use Personal AirPods with Work Apple TV

I work at a university and we have an AppleTV in one classroom and a Samsung TV with AirPlay in another. I host work days where I'm generally sitting for a few hours waiting for people to come in, so I want to stream shows to either TV and use my AirPods to listen. I'd rather not get out an extra-long HDMI cable.

So, how can I use either a MacBook or iOS device to accomplish this? Or can I temporarily share audio with my AirPods without pairing them to the Apple TV directly, as it uses a work Apple ID we use for all of our non-individual devices. Any thoughts? I have both AirPods Max and AirPods 2nd Gen, as well as an iPhone 14 Pro and a 2021 M1 Pro MacBook Pro.