### 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
GDScriptParser::DataType base_type = p_base;
	while (base_type.is_set()) {
		switch (base_type.kind) {
			case GDScriptParser::DataType::CLASS: {
				if (base_type.class_type) {
					if (base_type.class_type->has_member(p_symbol)) {
						r_result.type = ScriptLanguage::LOOKUP_RESULT_SCRIPT_LOCATION;
						r_result.location = base_type.class_type->get_member(p_symbol).get_line();
						r_result.class_path = base_type.script_path;
						Error err = OK;
						r_result.script = GDScriptCache::get_shallow_script(r_result.class_path, err);
						return err;
					}
					base_type = base_type.class_type->base_type;
				}
			} break;
			case GDScriptParser::DataType::SCRIPT: {
				Ref<Script> scr = base_type.script_type;
				if (scr.is_valid()) {