FP Camera vertical orbit around "poles" of an object

Hello, I'm a new user of Unity and have done several tutorials on standard first person games, but am now trying to do something different. I'm using PC keyboard controls.

I'd like a first person controlled camera which is confined to orbit a specific object at the centre of the scene. I'd like the left/right arrow keys to make the camera orbit horizontally around the "equator" of the object and the up/down arrow keys to make the camera orbit vertically around the "poles". I thought the easiest way to do this would be to just copy and tweak the FPSInputController, CharacterMotor and MouseLook scripts.

I got the horizontal orbit really easily by changing the MouseLook script to this:

 public Transform targetLookAt;
     
     void Update()
     {
         transform.LookAt(targetLookAt);
     }  

But the vertical one is escaping me. I tried changing line 11 in the FPSInputController script to say this:

 var directionVector = new Vector3(Input.GetAxis("Horizontal"), Input.GetAxis("Vertical"),0);

Thinking that it would use the up/down arrow input for the Y axis, but it doesn't work, I'm guessing because with the standard CharacterMotor the Y axis is set by jumping (or falling).

I'm sure there's a much easier way to do this than I'm trying to do it, probaby by writing a script from scratch rather than trying to hack the FPS ones about. Any advice on the best way to tackle would be much appreciated!

Thanks.