Bluetooth connection for PS4

Ps4 and the connected HDMI pico projector Joybee. The transmitter is connected to the handle of the PS4 go to the menu of the controller and validate total audio.
The transmitter is paired it to a pregnant EU BOOM which itself is master of a second enclosure EU BOOM, it can recover the sound in stereo bluetooth at the player.

in this case speaker on each side of the bed and projection on the wall does.

There's a slight shift: transition from the PS4 controller which transfers to the transmitter which processes and then sends it to the speaker, which deals also.
The shift is most important in my case if you connect the transmitter to the projector (processing time ???)

it should also run on PS3.

thank you for sony bluetooth with flanged on preventing connect to what you want.