### 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)

Dictionary dict = object->get_dict().duplicate();
	Variant key = dict.get_key_at_index(p_index);
	dict.erase(key);
	emit_changed(get_edited_property(), dict, "", false);
	update_property();
}
void EditorPropertyLocalizableString::update_property() {
	Variant updated_val = get_edited_property_value();
	if (updated_val.get_type() == Variant::NIL) {
		edit->set_text(TTR("Localizable String (Nil)")); 
		edit->set_pressed(false);
		if (container) {
			set_bottom_editor(nullptr);
			memdelete(container);
			button_add_item = nullptr;
			container = nullptr;
		}
		return;
	}
	Dictionary dict = updated_val;
	object->set_dict(dict);
	edit->set_text(vformat(TTR("Localizable String (size %d)"), dict.size()));
	bool unfolded = get_edited_object()->editor_is_section_unfolded(get_edited_property());
	if (edit->is_pressed() != unfolded) {
		edit->set_pressed(unfolded);
	}
	if (unfolded) {
		updating = true;
		if (!container) {
			container = memnew(MarginContainer);