"Is there something Weiner and company aren't telling us, or is this just a plot hole?"

It's possible that Dick entered the military in the first place to escape something other than a dreary farm life. Assuming Don's identity would let him escape the war and bury his past even further.

It could simply be that it was a 'heat of the moment' decision. He had just accidentally caused the death of the real Don by clumsily dropping his lighter in gasoline. That might be enough to make a man desparate to just get out of the war any way possible right then...

I agree with you that the benefits of the ID theft don't seem to outweigh the cost. There wouldn't be much story without it. Even some of the best TV/movie plots require at least a little willing suspension of disbelief.