That's really interesting. Obviously yours to do with as you wish but I want to share that my initial reaction was that you might want to consider leaving it as is.
That watch is more than a watch, like the kind that come and go in our collections, that's a watch with a story. And the story is "most true" in the state it is in.
Still, it's just a material object so I don't want to overstate it. I could see replacing the battery and even wearing it. Not sure how much cleaning up I would do; certainly not a restoration in my mind.
Thanks for sharing.