Fitness plus on external monitor without Wi-Fi or internet

Hi All,
I have an iPhone 11, a Watch 5 and, a Lightning-HDMI original adapter and a monitor with HDMI input.
I usually workout in a room where there isn't Wi-Fi connection and I need to download the video on my iPhone and stream offline.
I want to show the video on the external monitor because it's more easier to see on a bigger monitor than my iPhone but, if I connect the Lightning-HDMI adapter, the workout stops.
How can I solve?

I read in many forums that the streaming of Fitness+ is not available with the Lightning-HDMI adapter.
It's possibile to use a Apple TV WITHOUT Wi-Fi/Internet connection?
It's possible to link direclty the iPhone to Apple TV and stream Fitness+?

I want to be sure that it could work before buy another device.

Thanks for all the suggests
Fabio