I actually have two PS4 controllers, one older one and one with the light bar. Both of them are unable to connect via Bluetooth.
I don't have hid_sony or hid_playstation loaded, I tried manually loading them and I think that works but: https://i.imgur.com/VHbtW5l.png even though they show as loaded, I still experience the same problem: https://i.imgur.com/Z42WtKr.png I only have a bunch of unnamed devices when searching for new devices to connect to/pair with, and I can't pair with any of them. My headphones and Xbox Series X controller work perfectly.