我无法测试（或修改）您的jQuery代码，所以您必须自己处理，可能需要对它进行一些修改。我已经重新访问了您的所有代码（除了jQuery代码），一切都如您所料。
因此$package['destination']将有一个额外的'fias_code'条目。
注解代码：

// Add billing and shipping fields
add_filter( 'woocommerce_billing_fields' , 'custom_billing_fields' );
add_filter( 'woocommerce_shipping_fields' , 'custom_shipping_fields' );

function custom_shipping_fields( $fields ) {
    $fields['shipping_fias_code'] = array(
        'type' => 'text',
        'label' => __('FIAS', 'woocommerce'),
        'placeholder' => _x('Код', 'placeholder', 'woocommerce'),
        'required' => false,
        'class' => array('form-row-wide'),
        'clear' => true
    );

    return $fields;
}

function custom_billing_fields( $fields ) {
    $fields['billing_fias_code'] = array(
        'type' => 'text',
        'label' => __('FIAS', 'woocommerce'),
        'placeholder' => _x('Код', 'placeholder', 'woocommerce'),
        'required' => false,
        'class' => array('form-row-wide'),
        'clear' => true
    );
    return $fields;
}

// Ajax sender
add_action( 'wp_footer', 'checkout_send_fias_code_via_ajax_js' );
function checkout_send_fias_code_via_ajax_js() {
    if ( is_checkout() && ! is_wc_endpoint_url() ) :
    ?><script type="text/javascript">
    jQuery( function($){
        if (typeof wc_checkout_params === 'undefined')
            return false;

        // Function that send the Ajax request
        function sendAjaxRequest( value, fieldset = 'billing' ) {
            $.ajax({
                type: 'POST',
                url: wc_checkout_params.ajax_url,
                data: {
                    'action': 'fias_code',
                    'fias_code': value,
                    'fieldset' : fieldset
                },
                success: function (result) {
                    $(document.body).trigger('update_checkout'); // Update checkout processes
                    console.log( result ); // For testing (output data sent)
                }
            });
        }

        // Billing fias code change & input events
        $(document.body).on( 'change input', 'input[name=billing_fias_code]', function() {
            sendAjaxRequest( $(this).val() );
        });

        // Shipping fias code change & input events
        $(document.body).on( 'change input', 'input[name=shipping_fias_code]', function() {
            sendAjaxRequest( $(this).val(), 'shipping' );
        });
    });
    </script>
    <?php
    endif;
}

// The Wordpress Ajax PHP receiver (set data to a WC_Session variable)
add_action( 'wp_ajax_fias_code', 'set_fias_code_to_wc_session' );
add_action( 'wp_ajax_nopriv_fias_code', 'set_fias_code_to_wc_session' );
function set_fias_code_to_wc_session() {
    $field_key = 'fias_code';
    if ( isset($_POST[$field_key]) && isset($_POST['fieldset']) ){
        // Get data from custom session variable
        $values = (array) WC()->session->get($field_key);

        // Initializing when empty
        if( ! empty($values) ) {
            $values = array(
                'billing' => WC()->customer->get_meta('billing_'.$field_key),
                'shipping' => WC()->customer->get_meta('shipping_'.$field_key)
            );
        }

        // Sanitizing data sent
        $fieldset  = esc_attr($_POST['fieldset']);
        $fias_code = sanitize_text_field($_POST[$field_key]);

        // Set / udpate custom WC_Session variable
        $values[$fieldset] = $fias_code;
        WC()->session->set($field_key, wc_clean($values));

        // Send back to javascript the data received as an array (json encoded)
        echo json_encode(array($fieldset.'_'.$field_key => $fias_code));
        wp_die(); // always use die() or wp_die() at the end to avoird errors
    }
}

// Update checkout fields 'fias_code' values from custom WC_session variable
add_filter('woocommerce_checkout_get_value', 'update_fias_code_checkout_fields_values', 10, 2 );
function update_fias_code_checkout_fields_values( $value, $input ) {
    $field_key = 'fias_code';

    // Get data from custom session variable
    $values = (array) WC()->session->get($field_key);

    if ( ! empty($values) ) {
        if ( 'billing_'.$field_key === $input ) {
            $value = $values['billing'];
        }
        if ( 'shipping_'.$field_key === $input ) {
            $value = $values['shipping'];
        }
    }
    return $value;
}

// Add 'fias_code' data to destination shipping packages
add_filter( 'woocommerce_cart_shipping_packages', 'add_fias_code_to_destination_shipping_package' );
function add_fias_code_to_destination_shipping_package( $packages ) {
    $customer   = WC()->customer; // The WC_Customer Object

    // Get 'fias_code' data from customer meta data
    $main_key   = 'fias_code';
    $meta_value = $customer->get_meta('shipping_'.$main_key);
    $meta_value = empty($meta_value) ? $customer->get_meta('billing_'.$main_key) : $meta_value;

    // Get data from custom session variable
    $values = (array) WC()->session->get($main_key);

    if ( ! empty($values) ) {
        $session_value = $values['shipping'];

        if ( $session_value === $meta_value ) {
            $session_value = $values['billing'];

            if ( $session_value !== $meta_value ) {
                $meta_value = $values['billing'];
            }
        } else {
            $meta_value = $session_value;
        }
    }

    // Loop through shipping packages
    foreach ( $packages as $key => $package ) {
        // Set to destination package the "fias_code"
        $packages[$key]['destination'][$main_key] = $meta_value;
    }
    return $packages;
}

// Remove custom WC_Session variable once order has been created (before thankyou)
add_action( 'woocommerce_checkout_order_created', 'remove_fias_code_custom_wc_session_variable' );
function remove_fias_code_custom_wc_session_variable() {
    // Remove the custom WC_Session variable
    WC()->session->__unset('fias_code');
}

这段代码在functions.php文件的活动子主题（或活动主题）.测试和作品.

重要提示：

显示并保存fias_code checkout 字段：

我使用woocommerce_billing_fields和woocommerce_shipping_fields过滤器钩子代替woocommerce_checkout_fields，因为这样数据会保存为订单 meta数据和用户元数据，所以不再需要最后一个函数。
订单 meta键以下划线开头，就像所有的账单和发货单元数据一样。
字段将显示在我的帐户编辑地址...所以如果你想避免，你将需要添加一个条件，这两个相关的挂钩。

关于我的jQuery代码：

你最好把它复制到一个外部文件中，并以一种干净的WordPress方式注册/排队它，将输出限制在结帐页面。然后你将删除相关的操作钩子和钩子函数。

关于操作和筛选器挂接：

您将不得不更改所有add_action()和add_filter()为您的插件一样：

add_filter( 'woocommerce_billing_fields' , array($this, 'custom_billing_fields') );
add_filter( 'woocommerce_shipping_fields' , array($this, 'custom_shipping_fields') );
add_action( 'wp_footer', array($this, 'checkout_send_fias_code_via_ajax_js') );
add_action( 'wp_ajax_fias_code', array($this, 'set_fias_code_to_wc_session') );
add_action( 'wp_ajax_nopriv_fias_code', array($this, 'set_fias_code_to_wc_session') );
add_filter('woocommerce_checkout_get_value', array($this, 'update_fias_code_checkout_fields_values'), 10, 2 );
add_filter( 'woocommerce_cart_shipping_packages', array($this, 'add_fias_code_to_destination_shipping_package') );
add_action( 'woocommerce_checkout_order_created', array($this, 'remove_fias_code_custom_wc_session_variable') );