### 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)

value = dict.get_value_at_index(i + offset);
			EditorProperty *prop = memnew(EditorPropertyText);
			prop->set_object_and_property(object.ptr(), prop_name);
			int remove_index = 0;
			String cs = key.get_construct_string();
			prop->set_label(cs);
			prop->set_tooltip_text(cs);
			remove_index = i + offset;
			prop->set_selectable(false);
			prop->connect("property_changed", callable_mp(this, &EditorPropertyLocalizableString::_property_changed));
			prop->connect("object_id_selected", callable_mp(this, &EditorPropertyLocalizableString::_object_id_selected));
			HBoxContainer *hbox = memnew(HBoxContainer);
			property_vbox->add_child(hbox);
			hbox->add_child(prop);
			prop->set_h_size_flags(SIZE_EXPAND_FILL);
			Button *edit_btn = memnew(Button);
			edit_btn->set_icon(get_theme_icon(SNAME("Remove"), SNAME("EditorIcons")));
			hbox->add_child(edit_btn);
			edit_btn->connect("pressed", callable_mp(this, &EditorPropertyLocalizableString::_remove_item).bind(edit_btn, remove_index));
			prop->update_property();
		}
		if (page_index == max_page) {
			button_add_item = EditorInspector::create_inspector_action_button(TTR("Add Translation"));
			button_add_item->set_icon(get_theme_icon(SNAME("Add"), SNAME("EditorIcons")));
			button_add_item->connect("pressed", callable_mp(this, &EditorPropertyLocalizableString::_add_locale_popup));
			property_vbox->add_child(button_add_item);
		}
		updating = false;
	} else {
		if (container) {