In tandem with the input mapper the function, it may be better to register over-time events like movement and camera rotation in the _Process function. Instead using Input.GetVector("Left", "Right", "Down", "Up") to accumulate movement and rotation.
See how it’s done in the third-person shooter demo:
This code could probably be simplified a bit in 4.0 onwards, but the gist is that you need to register four new actions (such as look_up, look_down, look_left, look_right) and bind them to the right stick on controller input. You then add to the camera rotation in _process() (not _input() or _physics_process()).
Mouselook input remains entirely separate (and in _input()) as it doesn’t emit those input actions.