### Godot version

v4.0.alpha11.official [afdae67cc] (when the feature was added) and newer (still happens in RC6)

### System information

Windows 10

### Issue description

I'm unable to handle when the user sets an exported node property (this feature: https://github.com/godotengine/godot/pull/62185) from a setter in the editor (tool script).

I don't know if this is intentional or not, but for me it was unexpected. I had to revert back to `NodePath`s.

### Steps to reproduce

- Create a scene
- Attach a script to a node
- Add `@tool` at the top of the script
- Export a node property with a setter, preferably with an easy way to see if it executed:
	```
	@export var node:Node:
		set (mod_value):
			prints("node", mod_value)
			node = mod_value
	```
- Save, close and open again the scene to make sure the tool script is running in the editor
- Set the node property from the inspector
- Observe that the setter didn't run

The setter is not bypassed when running the game, it is only bypassed in the editor.

We can compare with a `NodePath` property, which setter is not bypassed regardless if it is in the editor or in game runtime.

### Minimal reproduction project

[nodepath.zip](https://github.com/godotengine/godot/files/10840411/nodepath.zip)

assign->set_icon(Ref<Texture2D>());
		assign->set_text(p);
		return;
	}
	Node *target_node = base_node->get_node(p);
	ERR_FAIL_COND(!target_node);
	if (String(target_node->get_name()).contains("@")) {
		assign->set_icon(Ref<Texture2D>());
		assign->set_text(p);
		return;
	}
	assign->set_text(target_node->get_name());
	assign->set_icon(EditorNode::get_singleton()->get_object_icon(target_node, "Node"));
}
void EditorPropertyNodePath::setup(const NodePath &p_base_hint, Vector<StringName> p_valid_types, bool p_use_path_from_scene_root, bool p_pointer_mode) {
	pointer_mode = p_pointer_mode;
	base_hint = p_base_hint;
	valid_types = p_valid_types;
	use_path_from_scene_root = p_use_path_from_scene_root;
}
void EditorPropertyNodePath::_notification(int p_what) {
	switch (p_what) {
		case NOTIFICATION_ENTER_TREE:
		case NOTIFICATION_THEME_CHANGED: {
			Ref<Texture2D> t = get_theme_icon(SNAME("Clear"), SNAME("EditorIcons"));
			clear->set_icon(t);
		} break;
	}
}
void EditorPropertyNodePath::_bind_methods() {