Dark Charm and Objective tokens
In short, NO.

My reasoning is based on the two pieces of information:

The Dark Charm card mentioned above has the following text (emphasis mine):


  Play this card on a hero at the start of your turn. The hero tests Willpower. If he passes, draw 1 Overlord card. If he fails, you may perform a move or attack action with that hero as if he were one of your monsters this turn. You cannot force him to suffer Fatigue or use a Potion, but you may force him to attack himself.


The only two actions that the overlord can take after using this ability is to move or attack.

If that isn't sufficient, trading is described as follows (emphasis mine):


  Heroes may give or receive Shop Item cards, Relic cards, and Search
  cards to or from each other during a move action. At any point during
  his move action, a hero may trade any number of Shop Item cards, Relic
  cards, or Search cards with an adjacent hero. Trading does not require an
  additional action or the expense of movement points.


There's nothing here stating that the hero can trade with an enemy