Is it permissible for an engaged man to pray to God to annul his engagement so that he may marry another woman he loves, given his fear of family problems?

There is no impediment to supplicating for the avoidance of an engagement without issues. The best approach is to perform Istikhara (prayer for guidance) according to the Hadith of Jabir ibn Abdullah, narrated by Al-Bukhari and others, which involves praying two rak'ahs and then reciting the well-known supplication for Istikhara.