BR-C1 Cannot connect to bluetooth headphones via Chromecast

Hey guys

My projector is connected to a Chromecast device and my audio comes out of a stereo box by an AUX cable (the AUX cable is connected to the projector and stereo).

I bought the Aukey BR-C1 so I could listen via my Bluetooth headphones (Sony MDR-XB950N1) and still watch my movies on the big screen.

I tried connecting the Aukey device to the projector (it has Audio Out) with the AUX cable and then connecting to my Mac with bluetooth–> No sound.
Then I tried connecting with the RCA audio cable and also connecting to my Mac with bluetooth–> No sound.
Unfortunately, the Aukey device doesn’t pair directly to the Bluetooth headphones.

Any ideas how to pair Aukey with the Chromecast or directly to my headphones?