Is it permissible for a husband not to spend the night with his disobedient wife (nashiz) who prevents him from seeing his family and children, expels him from the home, threatens to kill him, and verbally abuses him and his family, even though he is keen on the well-being of his children and provides for their material and psychological needs?

Your wife is rebellious (nashiz), and Allah has clarified how to deal with her in the Quran (An-Nisa:34). There is no sin upon you for spending the night with your second wife if the rebellious wife denies herself enjoyment, because her right to equal division (of nights) is forfeited due to her rebelliousness. You should continue to admonish and remind her, and seek help from someone she respects to reconcile. Divorce is not recommended unless the discord between you becomes severe and there is no longer hope for harmony and stability.