Good point!
Yes, I read that the learned and scholarly Chamberlain tended to embellish and romanticize his accounts after the war.
Apparently Chamberlain received many letters during his lifetime. From what I can ascertain, however, there is no evidence or confirmation of the existence of this particular letter. The publisher, Hearst, may well have published dramatized versions of Chamberlain’s recollections in a complete letter form to sell more publications.
I suspect, though, that there would be some basis of truth in any reported story, even if it was exaggerated.