Would the husband's public blessing of his son affect his marriage to a non-scriptural infidel woman in terms of the legitimacy of his second wife's marriage to him?

The marriage of your husband's son to a non-Kitābī (non-People of the Book) disbelieving woman is unanimously forbidden, and it is not permissible for your husband to pray for blessings for him, or to congratulate him on this marriage, or to participate in it. This is based on the Almighty's saying: "And He has already revealed to you in the Book that when you hear the verses of Allah being denied and ridiculed, do not sit with them until they enter into another conversation. Indeed, in that case, you would be like them. Indeed, Allah will gather the hypocrites and disbelievers in Hell all together." However, these actions do not take your husband out of Islam. It is obligatory to advise the young man and remind him of the gravity of this marriage, and that it is fornication, and to strive to dissuade him from it with gentleness and firmness, and to pray for his guidance.