What causes the chaotic trajectory of a balloon going flat?

Pointing out a single factor for the chaos is impossible in this case. It can be anything from vertical flow of surrounding air to surface deformities of the balloon membrane to even radiation pressure(this may be a small factor). The list may go on. Analyzing such a complex dynamical system without proper lab equipments is impossible. Even then I doubt if one can find the sole cause. Although, the main reasons may be:

1) non-uniformity near the nozzle of the balloon, where the flow supposedly becomes turbulent and therefore chaotic.

2)surrounding atmospheric phenomenon.

Chaos and irregular motion are not one and the same thing!

Chaotic motion is a precisely defined concept and every irregular motion that you see in a physical system need not be chaotic. However, chaotic motion always seems to be irregular. To be specific, chaotic motion refers to the deterministic motion that sensitively depends on the initial condition of the system. Also, the motion should satisfy things like "boundedness" and "topological mixing".

This means that if you have many obvious random variables in the system, you should not use the term "chaos" to describe the observed irregularities. For example, atomic transitions are irregular, but not chaotic. In the case of the balloon you are describing, we do not have (at least I do not know of) any model that is deterministic. If you have such a model, then one can try to find our whether it exhibits chaotic motion (at least for some ranges of its parameters) and only then you would be able to ask questions like which parameters cause chaos in the model and so on.

In fact, as I see it, the balloon's behavior, in this case, seems to be only a transient to me (because it stops after all the air has escaped) and hence cannot be chaotic. To understand more about the actual concept of chaos, you can try to read this.