I got it working before - BBC iPlayer app on iPhone, tap Airplay icon on a video and then choose Apple TV

It doesn’t seem to work via the Apple TV app (controlled via remote) = I believe because the app has barcoded DNS

But this approach isn’t Airplay mirroring (so it’s literally just using your phone as a remote to put it on Apple TV) so doesn’t stop you closing the app / using your phone etc.