FlyCam and LookAt problem

This is my first question here, so please bear with me :). I also did some searches and seemed to find some issues similar to my problem, but they unfortunately weren't close enough to help me. Or perhaps I didn't know the correct terminology to look up.

I found a pretty good FlyCam script (found here: http://wiki.unity3d.com/index.php/FlyCam_Extended) that I use to control my camera, and it works great. I more or less pulled a "script kiddie with this, but I was able to tweak it a bit for my needs, but here is the code for it:

 using System.Collections;
 using System.Collections.Generic;
 using UnityEngine;
 
 public class PlayerController : MonoBehaviour
 {
     /*
     EXTENDED FLYCAM
         Desi Quintans (CowfaceGames.com), 17 August 2012.
         Based on FlyThrough.js by Slin (http://wiki.unity3d.com/index.php/FlyThrough), 17 May 2011.
  
     LICENSE
         Free as in speech, and free as in beer.
  
     FEATURES
         WASD/Arrows:    Movement
                   Q:    Climb
                   E:    Drop
                       Shift:    Move faster
                     Control:    Move slower
                         End:    Toggle cursor locking to screen (you can also press Ctrl+P to toggle play mode on and off).
     */
 
     public static PlayerController instance;
     public float clickToSpeed = 0.5f;
     public bool clickToLocation;
     public Transform target;
 
     public float cameraSensitivity = 200;
     public float climbSpeed = 20;
     public float normalMoveSpeed = 100;
     public float slowMoveFactor = 0.25f;
     public float fastMoveFactor = 3;
 
     private float rotationX = 0.0f;
     private float rotationY = 0.0f;
 
     void Awake()
     {
         if (instance == null)
         {
             instance = this;
             //DontDestroyOnLoad(gameObject);
         }
         else if (instance != this)
         {
             Destroy(gameObject);
         }
     }
 
     void Start()
     {
         //Screen.lockCursor = true;
     }
 
     void Update()
     {
         if (Input.GetMouseButton(1))
         {
             rotationX += Input.GetAxis("Mouse X") * cameraSensitivity * Time.deltaTime;
             rotationY += Input.GetAxis("Mouse Y") * cameraSensitivity * Time.deltaTime;
             rotationY = Mathf.Clamp(rotationY, -90, 90);
 
             transform.localRotation = Quaternion.AngleAxis(rotationX, Vector3.up);
             transform.localRotation *= Quaternion.AngleAxis(rotationY, Vector3.left);
         }
         
 
         if (Input.GetKey(KeyCode.LeftShift) || Input.GetKey(KeyCode.RightShift))
         {
             transform.position += transform.forward * (normalMoveSpeed * fastMoveFactor) * Input.GetAxis("Vertical") * Time.deltaTime;
             transform.position += transform.right * (normalMoveSpeed * fastMoveFactor) * Input.GetAxis("Horizontal") * Time.deltaTime;
         }
         else if (Input.GetKey(KeyCode.LeftControl) || Input.GetKey(KeyCode.RightControl))
         {
             transform.position += transform.forward * (normalMoveSpeed * slowMoveFactor) * Input.GetAxis("Vertical") * Time.deltaTime;
             transform.position += transform.right * (normalMoveSpeed * slowMoveFactor) * Input.GetAxis("Horizontal") * Time.deltaTime;
         }
         else
         {
             transform.position += transform.forward * normalMoveSpeed * Input.GetAxis("Vertical") * Time.deltaTime;
             transform.position += transform.right * normalMoveSpeed * Input.GetAxis("Horizontal") * Time.deltaTime;
         }
 
 
         if (Input.GetKey(KeyCode.Q)) { transform.position += transform.up * climbSpeed * Time.deltaTime; }
         if (Input.GetKey(KeyCode.E)) { transform.position -= transform.up * climbSpeed * Time.deltaTime; }
 
         if (Input.GetKeyDown(KeyCode.End))
         {
             //Screen.lockCursor = (Screen.lockCursor == false) ? true : false;
         }
 
         GoToTarget();
     }
 
     private void GoToTarget()
     {
         if (clickToLocation)
         {
             transform.LookAt(target);
             transform.position = Vector3.Lerp(transform.position, target.position, clickToSpeed);
         }
 
         // Check to see if the player has gotten to the location.  If so, make clickToLocation false.
         if (transform.position == target.position)
         {
             clickToLocation = false;
         }
     }
 }

My issue comes in when I use transform.LookAt(target). I have an option for the user to click on an item from a panel and when they do, the camera looks towards the target, as it should, then Lerp is used to move them to it. All of that works great. The problem is that when I go to hold down the right-mouse button after this, the new direction I'm facing via LookAt isn't "known" by the variables connected to the mouse, so the camera quickly points back to the old direction that it "knows". So what I'd like to do is be able to update that information from what LookAt does. As I mentioned, I went very much "script kiddie" on this so I'm not entirely sure what is going on, but near as I can tell, I would think that if I can somehow get the info from LookAt into transform.localRotation, I'll be good to go:

 transform.localRotation = Quaternion.AngleAxis(rotationX, Vector3.up);
 transform.localRotation *= Quaternion.AngleAxis(rotationY, Vector3.left);

Am I in the ballpark on this? This feels right, but I just don't know how to get the information from the way the camera is facing from LookAt into transform.localRotation.

Thanks in advance!