The game only receives input events from the system, so it doesn't/know if it's from a Bluetooth controller or a wired controller, it's all handled the same. I'm not sure what's causing the lag, but it shouldn't be the game itself. Maybe you could try another game made in the Godot engine and see if you get the same issues? I had a look around, but I couldn't find any error/bug reports regarding Bluetooth latency issues