How can a girl be convinced to marry with Sharia-compliant evidence and convincing arguments, given her fear of problems between families? Is it permissible to pray for her to be a righteous wife for you, for you to be righteous for her, for you to be good for her and she good for you?

There is no harm in supplicating for ease in marrying your cousin (maternal aunt's daughter), while seeking out times when supplications are answered. You may also seek help from close relatives to persuade her. Marriage to relatives is permissible in Islamic law and has been mentioned in the Quran and Sunnah, and it does not necessarily lead to animosity. If her approval is not obtained, then entrust your affair to Allah, for He may grant you someone better than her, while being careful to choose a righteous wife.