"In other words, he would need to pick a Sam Brownback as his running mate."
He'd be more likely to pick Barney Frank.
To add one more thing here...
A strength that Giuliani has that is rarely mentioned is his ability to frame political issues in such a succinct and logical way that no matter what he says you find yourself nodding your head up and down.
The man would be a hell of a used car salesman.