Hi there! Unfortunately, PS4 systems don’t support Bluetooth headphones for audio at all. So unless Sony changes their minds…you’re stuck buying one of their official headsets, or a third party headset, and all of those use a USB dongle to achieve wireless connections.
It’s kind of silly since the consoles have native bluetooth support built in to connect the controllers, but that’s the world we’re living in.
I’m sorry I don’t have better news!