Other mistake: Toward the end of the movie when Bitsy is racing to get to the prison before the execution, she has car trouble. She tries the key again before giving up, grabbing the video tape and running down the road. This same "car trouble and desperate race" scene is shown at the opening of the movie, but Bitsy exits the car immediately after it stops. There is no delay to allow for her to try the key again or grab the video tape.
Other mistake: In the scene with the swimming pool full of mud and the kids playing on it, the sign on the entrance is supposed to be in Spanish, but most of the words are misspelled and no sentences make sense.
Answer: I believe that she was a plot device to make sure David Gale had nothing to lose by the end of the movie. She then may have felt guilty about casual sex, and tried to get back at Gale. She then later felt guilty about the whole charade, and wrote him a postcard admitting what she had done. Then the postcard is given to the ex-wife to show he was innocent of rape.
Richard Welty
David had asked Bitsy and, presumably, Dusty, to preserve his son's memory of him. So they/he/she forwarded the postcard from Berlin so his ex would know he was innocent of the rape charge. Along with money for his son.