### 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)

return;
		}
		float required_y_distance = 20.0f * EDSCALE;
		if (ABS(reorder_mouse_y_delta) > required_y_distance) {
			int direction = reorder_mouse_y_delta > 0.0f ? 1 : -1;
			reorder_mouse_y_delta -= required_y_distance * direction;
			reorder_to_index += direction;
			if ((direction < 0 && reorder_to_index % page_length == page_length - 1) || (direction > 0 && reorder_to_index % page_length == 0)) {
				
				_page_changed(page_index + direction);
			}
			property_vbox->move_child(reorder_selected_element_hbox, reorder_to_index % page_length);
			
			InspectorDock::get_inspector_singleton()->ensure_control_visible(reorder_selected_element_hbox);
		}
	}
}
void EditorPropertyArray::_reorder_button_down(int p_index) {
	if (is_read_only()) {
		return;
	}
	reorder_from_index = p_index;
	reorder_to_index = p_index;
	reorder_selected_element_hbox = Object::cast_to<HBoxContainer>(property_vbox->get_child(p_index % page_length));
	reorder_selected_button = Object::cast_to<Button>(reorder_selected_element_hbox->get_child(0));
	
	
	Input::get_singleton()->set_mouse_mode(Input::MOUSE_MODE_CAPTURED);
}
void EditorPropertyArray::_reorder_button_up() {