### 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)

int renamed_layer_index = -1;
	PopupMenu *layer_rename = nullptr;
	ConfirmationDialog *rename_dialog = nullptr;
	LineEdit *rename_dialog_text = nullptr;
	void _rename_pressed(int p_menu);
	void _rename_operation_confirm();
	void _update_hovered(const Vector2 &p_position);
	void _on_hover_exit();
	void _update_flag();
	Size2 get_grid_size() const;
protected:
	void _notification(int p_what);
	static void _bind_methods();
public:
	uint32_t value = 0;
	int layer_group_size = 0;
	int layer_count = 0;
	Vector<String> names;
	Vector<String> tooltips;
	void set_read_only(bool p_read_only);
	virtual Size2 get_minimum_size() const override;
	virtual String get_tooltip(const Point2 &p_pos) const override;
	void gui_input(const Ref<InputEvent> &p_ev) override;
	void set_flag(uint32_t p_flag);
	EditorPropertyLayersGrid();
};
class EditorPropertyLayers : public EditorProperty {
	GDCLASS(EditorPropertyLayers, EditorProperty);
public:
	enum LayerType {