Should I cover my bicycle overnight while bikepacking?
As Daniel indicates - overnight moisture on a well maintained bike isn't a concern. A little rain won't hurt. Others may disagree but I wouldn't get a cover.  
But, depending on how much rain there is and how long it lasts, when riding in the rain often the rain mixes with dirt/dust/sand and creates an abrasive liquid that can be hard on the drive train - and can even work it's way into bearings and cables.  
Fenders help keep the water mixed with dirt that's on the road off of you and the bike.  
I think of rain / water as a riding factor that increases how often I should do maintenance. 30+ days on the road should include at least drive train cleaning and lubrication.  
Here is a link to an article on bike prep for rain. The article links to washing and lubrication articles.