It is definitely an AirPod / AppleTV issue.
I have a Series 6 Apple Watch (cellular), a Gen1 pair of AirPods, a pair of AirPod Pros and a 2021 AppleTV. Whenever I try to do any workout on Fitness+ while using the AppleTV, my Apple Watch, and my AirPods (I've tried both models) then the Apple Watch disconnects from the AppleTV multiple times during a 30 minute workout. It's not even worth the effort to try the bluetooth reset option because the Apple Watch will just disconnect again within a few minutes.
As soon as I stop using either pair of AirPods, the Apple Watch and AppleTV have no issues. I just have to listen to the audio directly from the TV. As others have mentioned, I work out at 5am every day when my family is sleeping. Having my TV blaring the enthusiastic positivity of the Fitness+ team (motivating and life-affirming as they might be) has not made me a popular person in my household.
I would add that I have an older iPad Pro and I can use either set of AirPods and my Apple Watch with my iPad (I've also used my iPhone too) and there are no disconnect issues.
The problem only occurs when the AirPods are used with the AppleTV.
I am super bummed because I bought the AppleTV for the sole purpose of using it for Fitness+. As of now, it is an expensive paper weight.
One would think that Apple would quickly solve this issue because it affects those customers willing to buy headphones, a watch, a TV device, AND pay $10/mo for the Fitness+ subscription. The whole point of buying Apple products is to have an amazing experience if you buy into their entire ecosystem. The whole is greater than the sum of the parts. In this case, however...... not so much. The first post in this thread was Dec 15, 2020. It is now Jan 13, 2022. This is really sad.