I believe the PS Bluetooth is only for PS peripherals like DualSense & their own headphones. Third party headphones aren’t supported.
There’s some suggestion that purchasing a BT USB dongle is a workaround though.
Weird. It is a link to the Amazon site showing the dongle you require.
Search for PS5 wireless transmitter. Ensure it has the connect button. If they’re for the console then it should. It also needs to be audio. Adapter
This is the U.K. site so may need to check for the same or similar in your Country’s Amazon store.