### 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
for (const PropertyInfo &E : pinfo) {
							if (E.usage & (PROPERTY_USAGE_GROUP | PROPERTY_USAGE_CATEGORY)) {
								continue;
							}
							if (E.name.contains("/")) {
								continue;
							}
							int location = p_recursion_depth + _get_property_location(type, E.name);
							ScriptLanguage::CodeCompletionOption option(E.name, ScriptLanguage::CODE_COMPLETION_KIND_MEMBER, location);
							r_result.insert(option.display, option);
						}
						List<MethodInfo> signals;
						ClassDB::get_signal_list(type, &signals);
						for (const MethodInfo &E : signals) {
							int location = p_recursion_depth + _get_signal_location(type, StringName(E.name));
							ScriptLanguage::CodeCompletionOption option(E.name, ScriptLanguage::CODE_COMPLETION_KIND_SIGNAL, location);
							r_result.insert(option.display, option);
						}
					}
				}
				bool only_static = base_type.is_meta_type && !Engine::get_singleton()->has_singleton(type);
				List<MethodInfo> methods;
				ClassDB::get_method_list(type, &methods, false, true);
				for (const MethodInfo &E : methods) {
					if (only_static && (E.flags & METHOD_FLAG_STATIC) == 0) {
						continue;
					}
					if (E.name.begins_with("_")) {
						continue;
					}