### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
for (int i = 0; i < tool_menu->get_item_count(); i++) {
		if (tool_menu->get_item_id(i) != TOOLS_CUSTOM) {
			continue;
		}
		if (tool_menu->get_item_text(i) == p_name) {
			if (tool_menu->get_item_submenu(i) != "") {
				Node *n = tool_menu->get_node(tool_menu->get_item_submenu(i));
				tool_menu->remove_child(n);
				memdelete(n);
			}
			tool_menu->remove_item(i);
			tool_menu->reset_size();
			return;
		}
	}
}
PopupMenu *EditorNode::get_export_as_menu() {
	return export_as_menu;
}
void EditorNode::_dropped_files(const Vector<String> &p_files) {
	String to_path = ProjectSettings::get_singleton()->globalize_path(FileSystemDock::get_singleton()->get_current_directory());
	_add_dropped_files_recursive(p_files, to_path);
	EditorFileSystem::get_singleton()->scan_changes();
}
void EditorNode::_add_dropped_files_recursive(const Vector<String> &p_files, String to_path) {