How can I lawfully marry a second wife, knowing that my first wife suffers from multiple illnesses that cannot be legally proven, and that we live in a country that does not permit polygyny or customary marriage, and witnesses fear legal repercussions?

You are permitted to circumvent an oppressive law in your country in a manner that suits you and does not cause you harm. For instance, you may marry secretly without registering the marriage in court, provided it fulfills its Islamic conditions and pillars. You can also move to another country. With piety and adherence to the religion, you will find ease from Allah, as He says: "And whoever fears Allah - He will make for him a way out" and "And whoever fears Allah - He will make for him ease in his affair."