I'm sure someone else will give you a more accurate answer.  But at least
this is a start:
You've confused two episodes.  Twain and Howells did indeed get all messed
up trying to reach some ceremonial occasion outside Boston--a source of
frustration at the time, but later something they laughed about.

But the episode  with the sealskin coat came several years earlier, and the
offended lady was Mrs. Thomas Bailey Aldrich.

I'm nearly sure both stories are in Justin Kaplan's biography.  If Kaplan
doesn't include a picture of the sealskin coat, you can find one on p. 115
of the old picture biography MARK TWAIN HIMSELF, by Milton Meltzer.

Mark Coburn