You granted capabilities for access to the custom post type “Badges”. Admin menu “BadgeOS” may be protected by other user capabilities. Try to activate “Admin menu access” add-on included into URE Pro. Go to “User Role Editor” select “administrator” role and click “Admin menu” button for it. Find “BadgeOS” menu and look what capabilities protects it. Then you may decide if you will grant them to your client admin role or will not.