WordPress can’t find IXR_Client

You mean class-IRC.php or class-IXR.php?

Just include the files before you call IXR_Client. I use such code to call IXR_CLIENT and it worked on my site:

include_once( ABSPATH . WPINC . '/class-IXR.php' );
include_once( ABSPATH . WPINC . '/class-wp-http-ixr-client.php' );

$client = new WP_HTTP_IXR_CLIENT( 'http://othersite.com/xmlrpc.php' );