The key to the issue is in the first sentence of the FAM quote — it only becomes an issue in the US or for immigration if the intent is to practice polygamy in the US. That usually only has an effect if there is an attempt to immigrate the second spouse (e.g., immigrate spouse 1, divorce them, “remarry” spouse 2, immigrate them, all live happily together in US). No divorce of spouse 2 is required, but there should no longer be a spousal relationship. Technically, if that relationship with spouse 2 is maintained on trips outside the US, and that can be proven, there might be an issue. Just like “intent is decided at the border” for other issues, once the immigrant visa is issued and the person is admitted to the US, and naturalized, it becomes very difficult to prove the intent to practice polygamy in the US.
None of this is an issue for any biological children involved. If they are his biological children, regardless of whom the mother may be (former spouse, current spouse 2 with/without an ongoing spousal relationship, child from an adulterous relationship, born out-of-wedlock — whatever), he can petition for the child(ren) without repercussions.
OP, to answer your question, you will have to prove — as does any parent petitioning their child — the relationship between you and each child.