In Doom 3, how do I get my Bluetooth-connected game controller to work? This same controller, connected via USB, works fine.

It is successfully connected to my Windows 11-based PC...

I have it paired with Windows via the Pro Controller Input connector

It shows and works fine in joy.cpl

I have the controller configured in both Steam and X360CE

Thanks.

Doom 3 never had controller support on PC, and if you're playing the Steam version of it, you're likely using a Steam Input configuration that is mapping the keyboard and mouse inputs to your controller. There is a way however to get controller support with Doom 3 without using a program like Xpadder (which is obsolete compared to newer and FOSS alternatives like AntiMicroX) by running the game in the dhewm3 source port, which did get partial controller support a while back with its v1.5.3 update. I've used it myself and it pretty well after some of the bugs were ironed out with it, and it also supports many controllers used by SDL2 like Xbox Wireless Controllers for the Xbox One or Series X|S, DualShock 4, DualSense, and Switch Pro Controller.