### 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
return "";
}
void GDScriptLanguage::get_recognized_extensions(List<String> *p_extensions) const {
	p_extensions->push_back("gd");
}
void GDScriptLanguage::get_public_functions(List<MethodInfo> *p_functions) const {
	List<StringName> functions;
	GDScriptUtilityFunctions::get_function_list(&functions);
	for (const StringName &E : functions) {
		p_functions->push_back(GDScriptUtilityFunctions::get_function_info(E));
	}
	
	{
		MethodInfo mi;
		mi.name = "preload";
		mi.arguments.push_back(PropertyInfo(Variant::STRING, "path"));
		mi.return_val = PropertyInfo(Variant::OBJECT, "", PROPERTY_HINT_RESOURCE_TYPE, "Resource");
		p_functions->push_back(mi);
	}
	{
		MethodInfo mi;
		mi.name = "assert";
		mi.return_val.type = Variant::NIL;
		mi.arguments.push_back(PropertyInfo(Variant::BOOL, "condition"));
		mi.arguments.push_back(PropertyInfo(Variant::STRING, "message"));
		mi.default_arguments.push_back(String());
		p_functions->push_back(mi);
	}
}
void GDScriptLanguage::get_public_constants(List<Pair<String, Variant>> *p_constants) const {