Set the WordPress site url and home to your local server url
This is less a TIL and more a dirty hack, but whatever:
define('WP_SITEURL', 'http://' . $_SERVER['HTTP_HOST']);
define('WP_HOME', 'http://' . $_SERVER['HTTP_HOST']);
Put this in your wp-config.php
and you're golden.
Don't do this on a live server, though.