I also have the same problem.

Dualshock 4 Controller.
Windows 11 System
FFXIV played through its own launcher (non-Steam.)
And PC has innate Bluetooth connectivity.

The controller can't be found or recognized by the game itself. The controller works fine with other games.
Have tried resetting computer, controller, and redownloading and verifying file integrity as well.
Nothing has worked.

The only thing that works is having to plug in the controller through a USB port and it functions. Not ideal for me personally.

I hope that the problem is recognized and a fix to be found!