Well pretty much all we have is Choi's word, and the fact that he was at the same university where Funakoshi sometimes taught in that timeframe (so it would make sense)...but for that matter, for a lot of these early pioneers, all we have is what they later told biographers.
And in any case, it's kind of beside-the-point, isn't it? Your original question was, "Why is there no historical documentaton/verification of when General Choi trained Shotokan?" I've already explained when he would have trained (high school and college, by Choi's account) and why there would be no historical documentation (because people generally didn't write these kinds of things down back then, and even if they had, there was a war on).
Take my own story as an example...I originally studied "traditional taekwondo" (i.e., more karate-like) at a college club back in 1978 in a small college outside Charlotte, NC. We trained in a racquetball court, and everybody (except the instructor) wore white belts indefinitely because quite frankly nobody in the club was worrying about what color belt they had. I doubt you'd find "historical documentation/verification" of my training there, and hey -- we weren't even at war! There weren't that many of us in the club, and knowing us all by name was easy, so writing down that "Jim was in the taekwondo club" for posterity's sake would have been a little silly. So, all you have is my word, and the fact that I was at that college in the timeframe I said I was.
Because that's how things worked, back in the day.