Brianna would have been at least slightly injured when struck by the bison.
When Fergus is switching carrying the baby from his right arm to his left in order to help Marsali through labor, the pinky finger on the wooden hand moves and separates from the rest of the fingers. This confirms that the "wooden hand" is in fact a glove worn by the actor and painted so as to appear wooden.
The idea of sucking out the poison with one's mouth is potentially dangerous, while removing very little venom, as the venom does not rest in a pocket but immediately spreads through the tissue and into the blood stream.