Is it permissible for a wife to be examined at a free government hospital with the director's knowledge, despite having private health insurance that does not cover government hospitals, and the system does not permit it?

Your wife is not permitted to receive free medical treatment at the government hospital unless authorized by the empowered hospital director. If he is not empowered, then that is deception and treachery. We draw your attention to the fact that among health insurances, there is that which is Islamically permissible, which is cooperative, Takaful insurance, and among them there is that which is prohibited, which is commercial insurance based on gharar (excessive uncertainty) and gambling.