Kavanaugh has everything to gain and nothing to lose by asking - demanding - that the FBI open an investigation.

The FBI would question all of the other people there. They would compare his calendar with Judge's schedule and Blasey Ford's recommendation. They would produce a factual inquiry of the evidence that would exonerate him.

The FBI investigation would delay by a few weeks, and in exchange would clear the allegations and ensure his confirmation. And it would give him independent evidence to support his reputation.

He has everything to gain and nothing to lose... well, unless he's guilty. Then he would avoid it as much as possible.