### 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 layer_count = grid->layer_count;
	layers->clear();
	for (int i = 0; i < layer_count; i++) {
		const String name = get_layer_name(i);
		if (name.is_empty()) {
			continue;
		}
		layers->add_check_item(name, i);
		int idx = layers->get_item_index(i);
		layers->set_item_checked(idx, grid->value & (1 << i));
	}
	if (layers->get_item_count() == 0) {
		layers->add_item(TTR("No Named Layers"));
		layers->set_item_disabled(0, true);
	}
	layers->add_separator();
	layers->add_icon_item(get_theme_icon("Edit", "EditorIcons"), TTR("Edit Layer Names"), grid->layer_count);
	Rect2 gp = button->get_screen_rect();
	layers->reset_size();
	Vector2 popup_pos = gp.position - Vector2(layers->get_contents_minimum_size().x, 0);
	layers->set_position(popup_pos);
	layers->popup();
}
void EditorPropertyLayers::_menu_pressed(int p_menu) {