What should I do about my ex-husband's refusal to financially support his two daughters, and would I be sinning if I left their right to Allah, or must I sue him despite my aversion to courts and their lengthy proceedings?

Since the divorce has occurred, do not dwell on the past; instead, look to the future. Perhaps Allah willed good for you by separating you from this man. Your husband must give you your rights, including the deferred dowry and maintenance during the waiting period (iddah (definition)). He must also provide for his two daughters. The obligatory maintenance includes food, drink, clothing, and housing. Scholars have differed concerning the housing of the custodian. The preponderant opinion is that if a girl reaches the age of seven, she must be with her father until she is married, provided he has a wife or a suitable female custodian.