They're certainly behaving as if this us their only chance. My guess is Trump would refuse to nominate anyone who won't promise to protect him beforehand, maybe even on the record for his insurance, and aside from Kavanaugh that's proven to be beyond the pale for everyone else. So it's him or nobody.
Trump thinks he needs Kavanaugh to protect him.
The Republicans think they have to win because its a symbolic victory against the other side and all symbolic victories against the other side have to be won.