I don't think Gore would be seen as illegitimate by anyone who would have supported him in the first place. Furthermore, Hastert wasn't like Gingrich or Boehner, and the GOP is only a few short years away from seeing how Gingrich's shutdowns and impeachment attempts hurt them in 1996 and 1998. Even if they do try massive obstructionism, that has a good chance of backfiring, especially in the wake of the Clinton impeachment attempt. Without 9/11, the economy would be better than it was in OTL 2004, and in any case, it didn't hurt Bush all that much. I've already established up thread that I think "voter fatigue" is a myth, generally used as an excuse when the actual reason parties don't stay in power all that long is that eventually, something will go wrong and the party in power gets blamed for it.
I think that unless 9/11 occurs and Gore's conducting of the Afghanistan war turns into a disaster he'll get reelected. Incumbents generally only get defeated when there are recessions, major scandals, or foreign policy disasters.