What is the legitimate Sharia way out of the first wife’s refusal to accept the second wife’s continued presence, and her threatening to demand divorce or khul‘, given the husband’s need for the second wife in Western countries?

Divorce is in the hands of the man for a wisdom and the benefit of both parties, and there is no consideration for man-made laws that contradict this. If the man abuses his right and harms the woman, or if she dislikes him due to a deficiency in his religiosity and character, Islam has legislated for her to seek separation, even through Khul'. If you are compelled to remain with your foreign wife and the mother of your children, you can use Istir'a', which is depositing a testimony with witnesses (four according to some scholars and two according to the majority) that if you divorce your wife, you are compelled to do so and it is not an actual divorce. A condition for Istir'a' is that it must precede the act.