What is the ideal solution for the problem of a married man with children whose wife he constantly disagrees with, and she is unwilling to divorce and does not satisfy him sexually, while it is not customarily possible for him to marry a second wife, and he fears committing the forbidden?

Every household has its problems, and to solve them, one must understand their causes. These problems might stem from you due to poor treatment or lack of attention, so rectify your mistakes. Or the problems might be from the wife, in which case you must address them wisely, for the wife is fundamentally supposed to follow her husband. If compatibility is not achieved between the spouses, and problems persist, and the wife does not improve her behavior towards her husband, and he is unable to marry another, and fears falling into sin due to his desires not being met, then divorce is the solution. However, the wife must be given one last chance to reform.