Using bluetooth headphones while also connected to Digital AV Adapter

I run on a treadmill at the gym which has a Digital AV Adapter for the TV in front of me.

Normally I have to plug in my wired headphones to the jack supplied at the gym, which comes back from the TV.

I was wondering if I can use my bluetooth headphones (which easily pair with my phone) to listen to the audio from my phone, while it is connected to the Digital AV Adapter and playing the video on the TV?

This way I would have the video on the TV, but the audio through my wireless headphones.

NOTE: I noticed previously with my wired headphones, that if I connected the AV Adapter first, then plugged my headphones directly into my phone, I could get the audio directly from the phone (as opposed to from the jack that is plugged into the TV). So I was hoping this could also be done with the wireless headphones.

Thanks