There are some details around bluetooth and FluidNC that might be helpful to clarify.
FluidNC has separate Wifi and Bluetooth versions of the firmware. For direct connection, you can have Wifi or Bluetooth, not both. If you go with the Bluetooth firmware, the WebUI is gone and you need some separate device to control it. A bluetooth pendant could be created, or a phone app. Direct connection of a bluetooth controller like an Xbox controller would not work.
What you are essentially doing with a bluetooth controller like an Xbox controller is treating it like a keyboard. So, if you can connect it via Bluetooth to a computer or phone/tablet, you could use it to control it. I did this via my phone and WebUI v3. However, for that to work, the WebUI has to be active because that is what is translating the commands from the controller to something FluidNC understands. So, if the screen turns off, that stops working. Something similar should also be possible with a gcode sender. You’re making a bluetooth connection to the PC and the gcode sender translates that to FluidNC commands.