### 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 (int i = 0; i < p_class->members.size(); i++) {
		if (p_class->members[i].type == GDScriptParser::ClassNode::Member::FUNCTION) {
			const GDScriptParser::FunctionNode *function = p_class->members[i].function;
			r_funcs[function->start_line] = p_prefix.is_empty() ? String(function->identifier->name) : p_prefix + "." + String(function->identifier->name);
		} else if (p_class->members[i].type == GDScriptParser::ClassNode::Member::CLASS) {
			String new_prefix = p_class->members[i].m_class->identifier->name;
			get_function_names_recursively(p_class->members[i].m_class, p_prefix.is_empty() ? new_prefix : p_prefix + "." + new_prefix, r_funcs);
		}
	}
}
bool GDScriptLanguage::validate(const String &p_script, const String &p_path, List<String> *r_functions, List<ScriptLanguage::ScriptError> *r_errors, List<ScriptLanguage::Warning> *r_warnings, HashSet<int> *r_safe_lines) const {
	GDScriptParser parser;
	GDScriptAnalyzer analyzer(&parser);
	Error err = parser.parse(p_script, p_path, false);
	if (err == OK) {
		err = analyzer.analyze();
	}
#ifdef DEBUG_ENABLED
	if (r_warnings) {
		for (const GDScriptWarning &E : parser.get_warnings()) {
			const GDScriptWarning &warn = E;
			ScriptLanguage::Warning w;
			w.start_line = warn.start_line;