Activate an OnMouseDown event before clicking a UI button?

You could remove the button component from the images and instead attach a script like this:

 using UnityEngine;
 using UnityEngine.EventSystems;
 using System;
 
 public class UIEventHandler : MonoBehaviour, IPointerClickHandler
 {
     public event Action<string> onPointerClick;
     public void OnPointerClick(PointerEventData eventData)
     {
         if (onPointerClick != null) onPointerClick(gameObject.name);
     }
 }

Then in the script that has the actions for that scene have

 public UIEventHandler leftButton;
 public UIEventHandler rightButton;

and drag the buttons into the fields in the inspector. Have a method like,

 public void OnUIClick(string sender)
 {
     if (Physics.Raycast(Camera.main.ScreenPointToRay(Position), 
     out RaycastHit hit, detectDistance, layerMask) &&
     hit.collider.GetComponent<Interactable>() ! null)
     {
         //interact with the thing
     }
     else
     {
         if (sender == "LeftButton") //do button thing
         else if (sender == "RightButton") //do button thing
     }
 }

then in Start() subscribe to the events like this

 void Start()
 {
     leftButton.onPointerClick += OnUIClick;
     rightButton.onPointerClick += OnUIClick;
 }

You can add other kinds of interactions to the UIEventHandler class too, like IPointerEnter and IPointerExit for a visible button so you can decide to nope out of a visual highlight or not before it happens.