Smooth camera on moving platform

I have this script of camera controller which follow the player:

    public GameObject player;
    private float smoothness = 0.1f;
    private Vector3 offset;

	// Use this for initialization
	void Start () {
        offset = transform.position - player.transform.position;
	}
	
	// Update is called once per frame
	void FixedUpdate () {
        Vector3 desiredPosition = player.transform.position + offset;
        Vector3 smoothedPosition = Vector3.Lerp(transform.position, desiredPosition, smoothness);
        transform.position = smoothedPosition;
	}

Normally it goes nicely and very smooth. However when the player object boards a moving platform, the camera stutters back and forth a little which cause image tearing. What should I add to make it smooth?
Here is the script to hold the player object on a moving platform which I suspect as the cause of the issue:

    private Vector3 offset;
    private GameObject target = null;

    private void Start()
    {
        target = null;
    }

    private void OnTriggerStay(Collider other)
    {
        target = other.gameObject;
        offset = target.transform.position - transform.position;
    }

    private void OnTriggerExit(Collider other)
    {
        target = null;
    }

    private void LateUpdate()
    {
        if(target != null)
        {
            target.transform.position = transform.position + offset;
        }
    }

Thank you in advance!

Following @highpockets 's comment,: depending on the player controller script, it should be possible to move it even if it’s the child of the platform, but that’s not the issue here:

FixedUpdate is for physics, so the Camera script shouldn’t use it. Change it to LateUpdate and it will work.

(FixedUpdate is called at fixed intervals, independent of the framerate, so if your deltaTime is lower than the Time.fixedTimeStep (usually 0.016~0.02) it means update is called more often so Camera will “lag” behind because of the FixedUpdate being called less often than Update/LateUpdate)