### 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)

}
EditorPropertyPath::EditorPropertyPath() {
	HBoxContainer *path_hb = memnew(HBoxContainer);
	add_child(path_hb);
	path = memnew(LineEdit);
	SET_DRAG_FORWARDING_CDU(path, EditorPropertyPath);
	path->set_structured_text_bidi_override(TextServer::STRUCTURED_TEXT_FILE);
	path_hb->add_child(path);
	path->connect("text_submitted", callable_mp(this, &EditorPropertyPath::_path_selected));
	path->connect("focus_exited", callable_mp(this, &EditorPropertyPath::_path_focus_exited));
	path->set_h_size_flags(SIZE_EXPAND_FILL);
	path_edit = memnew(Button);
	path_edit->set_clip_text(true);
	path_hb->add_child(path_edit);
	add_focusable(path);
	dialog = nullptr;
	path_edit->connect("pressed", callable_mp(this, &EditorPropertyPath::_path_pressed));
}
void EditorPropertyClassName::_set_read_only(bool p_read_only) {
	property->set_disabled(p_read_only);
}
void EditorPropertyClassName::setup(const String &p_base_type, const String &p_selected_type) {
	base_type = p_base_type;
	dialog->set_base_type(base_type);
	selected_type = p_selected_type;
	property->set_text(selected_type);
}
void EditorPropertyClassName::update_property() {