This script automatically selects the first returned shipping option with live rates, and updates the total price of the checkout accordingly.
Paste the following code right before the closing </head> tag of your checkout template:
<script type="text/javascript"> jQuery(document).ready(function() { jQuery(document).ajaxComplete(function(event, request, settings) { if (settings.url.indexOf('GetShippingCost') != -1) { $("#fc_shipping_methods_inner input[type='radio']:first").prop("checked", true); FC.checkout.updatePrice(i); } }); }); </script>