PS4's bluetooth audio output is encrypted, so slamming in any old dongle is unlikely to work. That's why most bluetooth headsets can't pair/no sound output if you can pair.
Not what you're looking for, but you could cobble together a headset setup using optical out and have just chat play through a wireless headset. This will not let you chat back, however, using a PS Camera's mic, you can
Have not tested this myself, however, taking a cue from the PS Gold's setup: you might have some luck if you rock a DECT-based system. It would treat the USB connection as an audio-out, then the USB dongle (NOT a bluetooth dongle) would handle the connectivity. Again, NO IDEA if this will work, but if you've got Amazon Prime you could test it for free and return it if it's no bueno. Catch is: DECT headset systems are largely business-related, so you're starting at the $100 mark all the way to $3-400 for Plantronics gear.
Good luck. That's all the ideas I have.
My department uses the Logitech H820e