I finally solved it by using the newer 4.17.15.0 version of TocaEdit Xbox 360 Controller Emulator. This version must be launched separately and runs in the background while the Unreal Editor is running, so you no longer need to paste any files into the C:\Program Files\Epic Games\UE_5.4\Engine\Binaries\Win64 directory.
It seems that something in Unreal 5.4.2 completely broke the ability for the older x360ce emulator to work, because Unreal 5.3.2 works perfectly fine with the older emulator, as described here.