What is the Shar'i ruling regarding a son born from the marriage of a Muslim man to a Christian woman in America before a judge, knowing that the reason for the marriage was to obtain residency? Is this son considered illegitimate? And what is the optimal solution for raising this son Islamically, given the current wife’s (Muslim) refusal to accept him?

Your marriage to that woman is invalid because it was without a guardian (wali) or witnesses, and you must repent from it. The child resulting from this marriage is attributed to you, because he is not considered illegitimate as long as you both believed this marriage to be permissible. You must take this child into your care, and your wife has no right to object to that. You must also explain to her that he is your legitimate son.