### Godot version

4.1.1

### System information

windows 11

### Issue description

When registering a type with class_name and using that class name to declare a variable in code only, autocomplete works as expected anywhere in other nodes/scenes. However, if making a custom scene node that extends a script with the class_name, if that custom scene is instantiated in a different scene, even if the type is specified, it only shows the parent node's functions and not any of the extended functionality. I have made a basic project showing the issue

### Steps to reproduce

I have made a very simple project showing the issue. https://github.com/mmmmmmmmmmmmmmmmmmmdonuts/godot_4_autocomplete_issue essentially, the extended Node scene will only show the custom function when both the type is specified as well as the type afterwards following "as" 
e.g. `@onready var my_class_4: MyClass = $MyClass4 as MyClass`

### Minimal reproduction project

https://github.com/mmmmmmmmmmmmmmmmmmmdonuts/godot_4_autocomplete_issue
String class_name = p_info.class_name;
	bool is_enum = p_info.type == Variant::INT && p_info.usage & PROPERTY_USAGE_CLASS_IS_ENUM;
	
	if ((p_info.type == Variant::OBJECT || is_enum) && !class_name.is_empty()) {
		if (is_enum && CoreConstants::is_global_enum(p_info.class_name)) {
			return class_name;
		}
		return _trim_parent_class(class_name, p_base_class);
	} else if (p_info.type == Variant::ARRAY && p_info.hint == PROPERTY_HINT_ARRAY_TYPE && !p_info.hint_string.is_empty()) {
		return "Array[" + _trim_parent_class(p_info.hint_string, p_base_class) + "]";
	} else if (p_info.type == Variant::NIL) {
		if (p_is_arg || (p_info.usage & PROPERTY_USAGE_NIL_IS_VARIANT)) {
			return "Variant";
		} else {
			return "void";
		}
	}
	return Variant::get_type_name(p_info.type);
}
static String _make_arguments_hint(const MethodInfo &p_info, int p_arg_idx, bool p_is_annotation = false) {