I have a base site that I use to create new sites via New Blog Templates. We have some menus that include CSS classes and a few menu items that include descriptions (legacy data but causing issues nonetheless.)
The problem is that the database inserts are failing during the copy because the values are not SQL escaped nor are they using $wpdb->prepare. As a result, menu items are not be created in the wp_xx_posts table and the CSS classes for menu items aren't created in the wp_xx_postmeta table.
I have verified locally that updating two lines in copier.php do the trick (although you may prefer another solution.)
Line 988: $values = '("' . implode( '","', esc_sql($object) ) . '")';
Line 998: $values = '("' . implode( '","', esc_sql($object_meta) ) . '")';
Please update the plugin with a fix that addresses this issue.