When you use the controller via bluetooth with Steam, what type of controller is it registering as in the Steam controller configuration settings? Additionally what does it appear as in Device Manager when connected via bluetooth?
That’s what I figured. In bluetooth mode it’s operating as DirectInput essentially, but as a non-descript “generic” controller much like a dual shock 2 would in a PS1/2 to USB adapter. It’s likely the bluetooth connection that’s throwing D4 for a loop here. I bet if you were to add D4 as a Steam game and launch it through Steam you’d very likely have a working bluetooth connection in D4 thanks to Steam’s very versatile (and disabled friendly!) wrappers.
Is your bluetooth connection via a USB dongle or is it a built in bluetooth card on the motherboard? If it’s the former and you have a spare dongle from another bluetooth device you can try pairing to that and see if D4 works with it. Otherwise you’re likely going to have to go through the process of adding D4 as an external Steam game and using Steam’s wrapper to work with D4. I can’t guarantee that will work properly though given D4 has its own control over controllers and Steam’s control might interfere with that. You won’t know until you try.
Keep in mind that while we both know you aren’t trying to cheat or do anything nefarious, I don’t know how Warden would react to Steam hooking into the game like that. This is one of the annoyances of bluetooth controllers on a PC. Windows and most games aren’t designed for that type of connection with a controller since the games usually expect a USB connection so as to be able to properly identify all of the controller’s inputs.