Just for clarity ... you're using the Peloton App on the iPhone to AirPlay (not the Peloton Watch app), right?
Yeah, the Watch part of the setup is really just for the sensor / health reporting (when you start up, the Watch records a workout, and the iPhone by way of the Peloton app does too).
The Peloton app, for $13/month give you an App for: iPhone, Watch, iPad and Apple TV. When you install the Peloton App on the iPhone, you get the matching Watch app, and if you connect it to the Health App, then it gets all integrated, the Watch can be used for a hear rate sensor, and it sends that data (in real time) to the Peloton iPhone app. I used an extra heart rate sensor we had with the iPad, and that works great too, just a simple BT setup.
iPad w/ Peloton App
Works with Wahoo cadence sensor
Does
not work with Watch for HR
Does work with BT HR sensors
All workout results are in the Peloton App
iPhone + Apple Watch w/ Peloton App(s)
Works with Wahoo cadence sensor
Does work with Watch for HR
Does work with BT HR sensors
All workout results are in the Peloton App
All workout results are
also in Activity/Workout on iPhone