FPS Controller with relative X/Z movement

I'm making a game where the player can walk on walls. I'm trying to achieve this by making the player a child to whichever wall he/she is walking on. This way, all rotation, gravity and movement of the player assumes that the wall is actually the floor.

I'm using this modified FPS script:

 using UnityEngine;
 using System.Collections;
  
 [RequireComponent (typeof (Rigidbody))]
 [RequireComponent (typeof (CapsuleCollider))]
  
 public class FPSController : MonoBehaviour {
  
     public float speed = 10.0f;
     public float gravity = 10.0f;
     public float maxVelocityChange = 10.0f;
     public bool canJump = true;
     public float jumpHeight = 2.0f;
     public bool grounded = false;
  
  
  
     void Awake () {
         rigidbody.freezeRotation = true;
         rigidbody.useGravity = false;
     }
  
     void FixedUpdate () {
         if (grounded) {
             // Calculate how fast we should be moving
             Vector3 targetVelocity = new Vector3(Input.GetAxis("Horizontal"), 0, Input.GetAxis("Vertical"));
             targetVelocity = transform.TransformDirection(targetVelocity);
             targetVelocity *= speed;
  
             // Apply a force that attempts to reach our target velocity
             Vector3 velocity = rigidbody.velocity;
             Vector3 velocityChange = (targetVelocity - velocity);
             velocityChange.x = Mathf.Clamp(velocityChange.x, -maxVelocityChange, maxVelocityChange);
             velocityChange.z = Mathf.Clamp(velocityChange.z, -maxVelocityChange, maxVelocityChange);
             velocityChange.y = 0;
             rigidbody.AddForce(velocityChange, ForceMode.VelocityChange);
         }
  
         // We apply gravity manually for more tuning control
         //rigidbody.AddForce((this.transform.up * -gravity * rigidbody.mass));
         this.rigidbody.AddForce(-this.transform.up * 9.8f);
         grounded = false;
     }
  
     void OnCollisionStay () {
             grounded = true;    
     }
  
     float CalculateJumpVerticalSpeed () {
         // From the jump height and gravity we deduce the upwards speed 
         // for the character to reach at the apex.
         return Mathf.Sqrt(2 * jumpHeight * gravity);
     }
 }

The problem I'm having is that the script doesn't move the X and Z position of the player relative to the parent wall. If I try to move forward on a wall, nothing happens because the script views it as the Y axis, even though it's the player's relative Z.

How could I modify this so that the X and Z movement are relative?