Using Different nav_menu_css_class for different nav_walkers

The third parameter passed into nav_menu_css_classes should give you the information you need to sort out the different menus. Try:

function onpage_nav_menu_css_class($classes, $item, $args) {
  var_dump($args);
  return $classes;
}
add_filter('nav_menu_css_class', 'onpage_nav_menu_css_class', 1, 3);

Note: That will make a mess of your page. It is debugging/development only code.
I think that the theme_location should be sufficient for your purposes. Just check that value and conditionally apply your classes based on that.