$shipping_class_id = $product->get_shipping_class_id();
$shipping_class= $product->get_shipping_class();
$fee = 0;

if ($shipping_class_id) {
   $flat_rates = get_option("woocommerce_flat_rates");
   $fee = $flat_rates[$shipping_class]['cost'];
}

$flat_rate_settings = get_option("woocommerce_flat_rate_settings");
echo 'Shipping cost: ' . ($flat_rate_settings['cost_per_order'] + $fee);