Hi Ian,
Apologies as I may not understand your issue correctly. You are running the Fitness+ app from your iPhone and Airplay-ing it to your AppleTV? If that is the case I'd suggest running the Fitness+ app directly from your AppleTV.
-Launch Fitness+ on your AppleTV then on your Watch launch the Fitness app and press connect (the ability to connect may pop up automatically once you launch it on AppleTV. Then you'll just press connect).
If you are launching Fitness+ from your iPhone 11 and Airplay-ing it to the AppleTV, what happens when you try to AirPlay a 3 minute sample video from Safari or from AppeTV+? I suggest this test to see if the 60 second issue with Airplay is an issue limited to Fitness or a larger iPhone Airplay to AppleTV issue.