In a recent game, I tried to exchange an Overlord-as-Peasant for a Soldier and it didn't work. I toyed around, and the same problem exists for Band of Misfits and Page.
According to my read of the cards, this exchange should be possible. It also says so in the wiki: http://wiki.dominionstrategy.com/index.php/Band_of_misfits
Of course, it's only relevant in edge cases, and the no-exchange is probably better than the accidental "I couldn't tell which one was the real Peasant" exchange, but it still cost me some endgame Triumphal Arch points tonight, and I'm mad.
No, I'm not actually mad.