### 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)

set_bottom_editor(nullptr);
			memdelete(container);
			button_add_item = nullptr;
			container = nullptr;
		}
	}
}
void EditorPropertyLocalizableString::_object_id_selected(const StringName &p_property, ObjectID p_id) {
	emit_signal(SNAME("object_id_selected"), p_property, p_id);
}
void EditorPropertyLocalizableString::_notification(int p_what) {
	switch (p_what) {
		case NOTIFICATION_THEME_CHANGED:
		case NOTIFICATION_ENTER_TREE: {
			if (button_add_item) {
				button_add_item->set_icon(get_theme_icon(SNAME("Add"), SNAME("EditorIcons")));
			}
		} break;
	}
}
void EditorPropertyLocalizableString::_edit_pressed() {
	Variant prop_val = get_edited_property_value();
	if (prop_val.get_type() == Variant::NIL && edit->is_pressed()) {
		VariantInternal::initialize(&prop_val, Variant::DICTIONARY);
		get_edited_object()->set(get_edited_property(), prop_val);
	}
	get_edited_object()->editor_set_section_unfold(get_edited_property(), edit->is_pressed());
	update_property();
}
void EditorPropertyLocalizableString::_page_changed(int p_page) {