How can I add text to a single nav menu item?

You can try the wp_nav_menu_objects filter. This seems to work, but not sure what other side-effects it may have, you’ll want to test thoroughly.

function wpa_filter_nav_menu_objects( $items ){
    foreach( $items as $item ){
        if( 'Season' == $item->title ){
            $item->title = get_option('current_season') . ' Season';
        }
    }
    return $items;
}
add_filter( 'wp_nav_menu_objects', 'wpa_filter_nav_menu_objects' );