If he wanted to and won the Republican Party's support, I think he'd get elected. The question is whether he would. Totally aside from the personal exhaustion others have pointed out, there was a rather strong tradition dating back to Martin Van Buren of Presidents serving only one term. Lincoln himself had participated in a similar term-limits deal which led him to step down after his single term in the House.another question pertaining to TTL, same time period: does anyone think Lincoln would go on to serve two full terms ITTL if the war ended earlier, or would he conceivably fall victim to some other form of foul play? (personally, i'm convinced that he would be re-elected since he successfully ended the war, and sooner ITTL than IOTL)
Also, he had strong opposition even inside the Republican Party, both from people who thought him too conservative (such as the faction which would almost nominate Fremont against him in OTL 1864) and those who thought him too radical. It's very possible he could surmount this opposition if he was determined to run - note how he brought his cabinet behind him iOTL. Of course, his very able 1860 campaign manager David Davis was nominated for the Supreme Court in October 1862 (I think that'd happen despite any military PoD.)
So, it could go either way politically. However, I agree that Lincoln wouldn't want to run unless he thinks the country needs him for Reconstruction. Whether that's the case depends mostly on your timeline.
To my surprise, Berkeley County was actually represented in the First Wheeling Convention. But aside from that, none of those three counties were represented in either Wheeling Convention or Restored Virginia's legislature, nor did they vote in the statehood referendum. I don't think they'll join Kanawah ITTL.Possibly... but the area had been under the civil jurisdiction of the Restored Government of Virginia, like the rest of the west, and participated, more or less, in the constitutional convention and statehood referendums.