If they don't have the votes to confirm his nomination now, it is unlikely that they will have them after November. The people considered mostly likely to vote against him are not up for election. Jeff Flake and Bob Corker are retiring. Susan Collins (2020) and Lisa Murkowksi (2022) aren't up in 2018.
If Brett Kavanaugh withdraws or fails confirmation, they need several months to vet the replacement. If that happens now, they still have time this year. If they wait until after the election, it's unlikely they could vote Kavanaugh down and then replace him as nominee before the new term. This is a strong reason to make a decision now.
If a replacement for Anthony Kennedy isn't confirmed by October, the Supreme Court may have to start its term with only eight members. This makes it harder to rule on contested issues. Things that might be 5-4 with Kavanaugh in the majority become 4-4. This means that they have to rehear a bunch of cases so as to come to a real decision. Which means that other cases that they should consider will not get heard.
All that said, there is nothing procedurally preventing the lame duck Senate from confirming. If nothing else, so long as Republicans have the majority, they could change the rules. That doesn't make that a desirable outcome though. Beyond any procedural issues, they want to confirm a replacement now, so the Supreme Court can start its session with a full membership to hear arguments.