Here’s the thing. There is no reason at all to presume that the Pauline Privilege might have been illicit. Please explain why you have said twice now that it might have been because I don’t understand that part. I could be missing something but I just don’t see it.
She was not free to marry (in the eyes of the Church) until after PP dissolved the first marriage; however the 2nd marriage occured before PP was granted. The fact that the second marriage occured while the wife was still bound by the first marriage means that the second marriage no longer enjoys the favor of the law–because something objective is obviously present which prevented the marriage from happening in the first place. Any marriage enjoys the favor of the law, but only until something to the contrary is proven. A simple admission by one spouse that this is a second marriage is proof enough.
When an annulment is sought on grounds of “lack of consent” or “lack of form” or other criteria, it is up to the competent authority to decree that it was invalid–no one else can make this determination. However in the case of a person marrying for the second time, while the first spouse is still alive, the Church treats as a given that this second marriage attempt is invalid. If a 3rd marriage is sought, then a formal decree of nullity would be necessary but that’s a whole different topic and not germain to this one.