What are the strong proofs that the Qur’an is complete, not ناقص (incomplete), and not محرف (corrupted), so that the questioner may be reassured and invite his Christian friend to Islam?

Your friend is correct in his belief regarding the distortion (tahrif) of the Torah and the Gospel. Evidence for this, along with the affirmation and authenticity of the Quran, has been previously detailed in Fatwas numbers: 45916 and 34699. Fatwa number: 32971 addresses those who, claiming to be Muslims, believe in the distortion of the Quran. You can find some rational and historical proofs for the authenticity and preservation of the Holy Quran in Fatwas: 33907 and 39898. For further benefit, you may refer to Fatwas: 63873, 37648, 41086, 63965, 58131, 75230, 77825, and 41069. There are also useful books such as: "Refutation of the Orientalists' Claim of Quranic Distortion" by Dr. Ahmed Mu'adh Alwan Haqee, "Casting Aspersions on the Holy Quran and Responding to Those Who Do So" by Dr. Abd al-Muhsin bin Zayn al-Mutairi, and "The Christian Missionary Raid on the Authenticity of the Holy Quran" by Dr. Abd al-Radi Muhammad Abd al-Muhsin. It is also advisable to pay attention to books on the scientific miracles in the Holy Quran. Furthermore, one must be cautious about engaging in discussions with people of misguidance unless fully equipped with responses.