How can filial piety be reconciled with not wronging one's wife and son, given the mother's insistence on expelling the wife and son and having the apartment registered in her name in exchange for her رضا (contentment), and after the disputes that arose due to the married sister residing in the brother's house?

The most important thing we advise is wisdom in reconciling the rights of your family and your wife. Your mother does not have the right to demand that the apartment be registered in her name except with your consent, and you are not obliged to obey her in this, nor in her demand for your wife's divorce. Your wife has the right to an independent dwelling where no relatives live with her. Strive to mend your relationship with your family and maintain ties with them even if they cut you off.