That makes the timing awfully interesting (and makes Republican complaints about Democratic delays to the process a little easier to understand). If Kavanaugh were to withdraw his name today, and Trump were to nominate someone else in his place tomorrow, the GOP might be able to confirm the replacement before the midterms — but the timing would be tight and would require a faster confirmation process than for any current member of the Supreme Court.
The lame-duck session would be a safer bet, but it’s not without risk for the GOP. One problem is that they might lose the Senate — to repeat ourselves, there’s about a 30 percent chance of this. Because the Senate is a much heavier lift for Democrats than the House, in the scenarios where the GOP loses the Senate, they’d probably also lose the House by wide margin; in our simulations, Republicans lose an average of about 50 (!!) House seats in scenarios where they also lose the Senate. The House doesn’t have any say in the Supreme Court nomination process, but would Republicans really want to push forward a nomination after losing by such a landslide margin?
My guess is probably yes — a Supreme Court seat really is that important to them. But the politics are uncertain; there aren’t really a lot of recent precedents for a party taking such significant action during the lame duck session.