### Godot version

v4.2.dev.custom_build [0ca854232]

### System information

Pop Os 22.04 lts 12th Gen Intel® Core™ i7-12700H × 20 NVIDIA RTX A2000 8GB 32.0GB  ram

### Issue description

A node that has a script attached to it and it has a couple of child nodes (type of a child node doesn't matter). In the script you have an export variable of type Node. You save the scene into a file and you add it in level scene. When you start to duplicate that scene in the level and you'll hover over exported variable to show a node path, it will show an incorrect path. 


https://github.com/godotengine/godot/assets/33091666/75088d1b-c014-41e0-bae3-46610eccbad1



### Steps to reproduce

1. create a scene with a node and a child node.
2. attach a script to the parent node and create an export variable of type Node
3. assign child node to that exported variable through the editor
4. save the scene and create a level scene where you are going to add just saved scene.
5. _Possibly requires a reboot of the engine before seeing the bug_
6. start duplicating nodes in the level scene

### Minimal reproduction project

[NodeTest.zip](https://github.com/godotengine/godot/files/12781547/NodeTest.zip)

ERR_FAIL_COND_MSG(data.inside_tree && !Thread::is_main_thread(), "Moving child node positions inside the SceneTree is only allowed from the main thread. Use call_deferred(\"move_child\",child,index).");
	ERR_FAIL_NULL(p_child);
	ERR_FAIL_COND_MSG(p_child->data.parent != this, "Child is not a child of this node.");
	_update_children_cache();
	
	if (p_child->data.internal_mode == INTERNAL_MODE_FRONT) {
		if (p_index < 0) {
			p_index += data.internal_children_front_count_cache;
		}
		ERR_FAIL_INDEX_MSG(p_index, data.internal_children_front_count_cache, vformat("Invalid new child index: %d. Child is internal.", p_index));
		_move_child(p_child, p_index);
	} else if (p_child->data.internal_mode == INTERNAL_MODE_BACK) {
		if (p_index < 0) {
			p_index += data.internal_children_back_count_cache;
		}
		ERR_FAIL_INDEX_MSG(p_index, data.internal_children_back_count_cache, vformat("Invalid new child index: %d. Child is internal.", p_index));
		_move_child(p_child, (int)data.children_cache.size() - data.internal_children_back_count_cache + p_index);
	} else {
		if (p_index < 0) {
			p_index += get_child_count(false);
		}
		ERR_FAIL_INDEX_MSG(p_index, (int)data.children_cache.size() + 1 - data.internal_children_front_count_cache - data.internal_children_back_count_cache, vformat("Invalid new child index: %d.", p_index));
		_move_child(p_child, p_index + data.internal_children_front_count_cache);
	}
}
void Node::_move_child(Node *p_child, int p_index, bool p_ignore_end) {