I don’t think this is so complicated bernoullis principle. I think it may just be Newtons Laws at work.
From observing the motion of the balloon, you see that the greatest force upward is at the edges, while the least force upward is in the center. The balloon is more dense than air and the bouyant force is not enough to keep it suspended so it falls normally. If we were to use net force to find the required force upward to keep the balloon suspended it would be equal to the weight of the balloon downward. Therefore, towards the center of the fan, the upward force is nearly equal to the weight.
However, towards the outside of the fan the force upward is greater than the weight and movew the balloon upward. This explains why it rises and falls.
However, the spin is created by Bernoulli’s Principle and Efflux flow. The balloon, a round object, gets angular momentum from the force exerted upward because of the efflux flow around the balloon. The linear and angular momentums cause it to spin and then go the opposite direction, in short. I’m sure you could probably work out the actual equations if you wanted. :o