Are you sure you're running it in exclusive mode If you don't run it in exclusive mode, windows detects both an xbox controller and a generic wireless controller, leading to some problems in games. It has rumble when paired on Big Picture locally, just no rumble when paired via the Shield. I use InputMapper (which is what DS4windows switched to if I recall correctly) with my DS4 and don't have any problems (wireless or wired). Outside of the game I can see my comp responding to the inputs, but inside. But for some reason the DS4 wont work unless its plugged in via cable.
My fiance uses an Xbox controller and the game responds to that, and i even got a nintendo switch pro controller to work.
Im using a bluetooth dongle and everything connects fine. The only real issue is that it doesn't have Rumble this way. I cant figure out why my DS4 isnt working via bluetooth on my PC. I have a DS4 paired wirelessly with my NVIDIA Shield and it is recognized as the "Wireless Gamepad" generically in Shield, and Steam Big Picture notices it as a Xbox 360 Controller. Bluetooth 4.0 (via an adapter like this or built in pc). But I did notice that the "Steam" GameStream app (which basically just launches Big Picture Mode, very smiliary to the Steam Link) has great controller support. DO NOT USE DS4 InputMapper keyboard mapping does not work with InputMapper. So is Sony going to be releasing official.
then probably your BT dongle is not compatible. Also, check that your dongle is connected to a proper USB 2.0 port.
Some background: I have games across various platforms (Xbox Game Pass Ultimate, Steam, Epic Games, Amazon Games, etc) and I've found that with GameStream the controller support was very finnicky. Youll need a micro USB to USB tethered connection and a program called Inputmapper. So try removing everything but your BT dongle. I have the DS4 working great on my Desktop now, but I'm having some slight issues with GameStream. Any version of DS4Windows distributed on Uptodown is completely virus-free and free to download at no cost. Download rollbacks of DS4Windows for Windows. The big reason I actually wanted to do this is because I've been trying to setup GameStream with my NVIDIA Shield. It includes all the file versions available to download off Uptodown for that app.