### 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
#ifdef TOOLS_ENABLED
	bool th = EditorSettings::get_singleton()->get_setting("text_editor/completion/add_type_hints");
#else
	bool th = false;
#endif
	String s = "func " + p_name + "(";
	if (p_args.size()) {
		for (int i = 0; i < p_args.size(); i++) {
			if (i > 0) {
				s += ", ";
			}
			s += p_args[i].get_slice(":", 0);
			if (th) {
				String type = p_args[i].get_slice(":", 1);
				if (!type.is_empty()) {
					s += ": " + type;
				}
			}
		}
	}
	s += String(")") + (th ? " -> void" : "") + ":\n" + _get_indentation() + "pass # Replace with function body.\n";
	return s;
}
#ifdef TOOLS_ENABLED
#define COMPLETION_RECURSION_LIMIT 200
struct GDScriptCompletionIdentifier {