So, I installed BthPS3 and it worked fine, but I noticed my DualShock 4 (the new CUH-ZCT2 model) can only connect properly during pairing. I know from what I read on the BthPS3 research thread that when a DualShock 4 tries to connect while BthPS3 is installed, the first connection will be refused and then I have a 10 seconds window to connect the controller again without BthPS3 interfering, but that isn't working here.
What happens instead is that during that 10 seconds interval the DualShock 4 will connect and turn the light bar a solid white, Windows will play the "device connected" sound but after 5 to 10 seconds, the controller will turn itself off and windows will play the "device disconnected" sound.
The only way to use the controller is to remove the device from Windows then pair again by holding PS + Share. After the pairing is finished, the DualShock 4 will remain connected and will work fine until you disconnect it. To use it again, I need to re-pair the controller. Uninstalling BthPS3 instantly fixes this reconnection issue.
If it helps, my Bluetooth adapter is an Orico BTA-403 (VID 0A12, PID 0001, Rev 8891) and I'm running Windows 10 v1909. I wanted to provide some kind of log file but AFAIK BthPS3 doesn't have any...