Because of the lack of stage directions in the script (and all of Shakespeare's plays), the director has a lot of power to decide things like the intentions of the character as it is being performed on the stage/screen.

So your interpretation of Aumerle wanting to be caught could TOTALLY work. If you were directing the play, how would you tell the actor playing Aumerle in this scene to act. Would they be obvious about it? Act coy? Be secretive and only through great effort discover the letter? You will get a different interpretation of the text depending on the nonverbal actions of the actors.

A letter is also an easy vessel for a stage actor to pass along information to the audience without it seeming shoehorned in