### 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)

if (!scene_tree) {
		scene_tree = memnew(SceneTreeDialog);
		scene_tree->get_scene_tree()->set_show_enabled_subscene(true);
		scene_tree->get_scene_tree()->set_valid_types(valid_types);
		add_child(scene_tree);
		scene_tree->connect("selected", callable_mp(this, &EditorPropertyNodePath::_node_selected));
	}
	scene_tree->popup_scenetree_dialog();
}
void EditorPropertyNodePath::_node_clear() {
	if (pointer_mode) {
		emit_changed(_get_meta_pointer_property(), NodePath());
	} else {
		emit_changed(get_edited_property(), NodePath());
	}
	update_property();
}
bool EditorPropertyNodePath::can_drop_data_fw(const Point2 &p_point, const Variant &p_data, Control *p_from) const {
	return !is_read_only() && is_drop_valid(p_data);
}
void EditorPropertyNodePath::drop_data_fw(const Point2 &p_point, const Variant &p_data, Control *p_from) {
	ERR_FAIL_COND(!is_drop_valid(p_data));
	Dictionary data_dict = p_data;
	Array nodes = data_dict["nodes"];
	Node *node = get_tree()->get_edited_scene_root()->get_node(nodes[0]);
	if (node) {
		_node_selected(node->get_path());
	}
}
bool EditorPropertyNodePath::is_drop_valid(const Dictionary &p_drag_data) const {