This code is untested and may need some tweaking, but it can be the correct path to make your external API calls work.

We can try to use the WC_Session variable to set the shipping fee we want to pass:

function handle_api_response( $rates ) {

    // 在这里编写API调用代码.....
 
    if ($response_code === 200){
        $response_data = json_decode($response, true);
        $cost = $response_data['shipments'];
         
        if ( $cost > 0 ) {
            // 将费用设置为WC Session变量
            WC()->session->set('shipment_cost', floatval($cost));

            // 尝试触发“更新结账”Ajax事件
            ?><script>jQuery('body').trigger('update_checkout');</script><?php
        }
    }
}

Then we can call the WC_Session variable in your hook function:

add_filter('woocommerce_package_rates', 'update_shipping_costs', 10, 2);
function update_shipping_costs( $rates, $package ) {
    foreach ( $rates as $rate_id => $rate ) {
        // 从Session变量中获取新的费用
        $new_cost = WC()->session->get('shipment_cost');

        if( isset($rate->cost) && $rate->cost > 0 && $new_cost > 0 ) {
            $rates[ $rate_id ]->cost = $new_cost; // 设置新的费用
        }
    }
    return $rates;
}

However, we need something else to refresh the cached shipping method in order for it to take effect:

add_action('woocommerce_checkout_update_order_review', 'refresh_shipping_methods');
function refresh_shipping_methods( $post_data ){
    $bool = true;

    // 我们检查Session变量
    if ( WC()->session->get('shipment_cost') > 0 ) {
        $bool = false;
    }

    // 与运输方法一起使用时是必需的
    foreach ( WC()->cart->get_shipping_packages() as $package_key => $package ){
        WC()->session->set( 'shipping_for_package_' . $package_key, $bool );
    }
    WC()->cart->calculate_shipping();
}

Finally, we unset the WC Session variable (on checkout first load and thank you page) :

add_action('wp_footer', 'reset_wc_session_variable');
function reset_wc_session_variable() {
    if (is_checkout() && WC()->session->get('shipment_cost') > 0) {
        WC()->session->__unset('shipment_cost');
    }
}

Put the code into your child theme's functions.php file (or into a plugin). It may work.

Related: Remove shipping costs after checking custom checkbox in WooCommerce checkout