How can the Quranic command to ignore the ignorant be reconciled with the prophets' persistence in inviting their people, despite their rejection of the truth, as seen in the story of Noah (peace be upon him)?

There is no contradiction between ignoring the ignorant and calling to Allah. The verse from Surah (definition) Al-A'raf: ﴿خُذِ الْعَفْوَ وَأْمُرْ بِالْعُرْفِ وَأَعْرِضْ عَنِ الْجَاهِلِينَ﴾ [“Show forgiveness, enjoin what is good, and turn away from the ignorant”] combines all good qualities. “Turning away from the ignorant” means showing forbearance and patience towards them, and keeping oneself pure from associating with the foolish and disputing with the stubborn. It does not mean abandoning the call to Allah or teaching those who are ignorant, but rather turning away from the ignorance and harm of the ignorant without severing ties or blaming. This verse is definitive and serves as a discipline for the Prophet, peace and blessings be upon him, and the Muslim community, to endure those who wrong them. It does not mean turning away from someone who is ignorant of Allah’s rightful due. The ignorant are everyone who is ignorant, and Allah has commanded to protect oneself from confronting them in their foolishness, while still obligating to denounce their wrongdoings.