A toilet will work just fine without a vent...and, in your situation, it wouldn't affect any of the pre-existing devices, because they all have vents. Most inspectors won't approve an AAV unless it is the best solution of a bad situation. If possible, you might attach the new sink's vent to one of the others to the right in your diagram. It would need to be made at 42" or higher above the floor and there are limits on how many turns it can take plus, it must still be sloped back to the drain.
An AAV only lets air in. With it in that location, when you flushed the toilet, there's a pressure wave in front of the wave as the stuff flows down the pipe. That would want to push stuff to the point of least resistance. It probably wouldn't spit out the new sink, but since the AAV can't let that pressure wave out, it likely would rock the water in the trap.