I tried copying the responsive_menu table from another working installation and now more of the admin is loading, but still getting an error in the search box section:
Catchable fatal error: Argument 1 passed to ResponsiveMenu\Form\Colour::render() must be an instance of ResponsiveMenu\Models\Option, null given, called in /public_html/wp-content/plugins/responsive-menu/src/app/ViewModels/Components/Admin/Boxes.php on line 66 and defined in /public_html/wp-content/plugins/responsive-menu/src/app/Form/Colour.php on line 10
So it looks like the problem is due to the database not updating properly. Is there another table that the plugin needs?