Is it obligatory to inform the teacher about correcting recitation from the Mus'haf without her knowledge during the ختم القرآن (completion of the Qur'an), or is it sufficient to re-recite the specific surah to her?

Reciting the Quran has great merit, for every letter counts as ten good deeds, according to the Prophet, peace and blessings be upon him, who said: "Whoever recites a letter from the Book of Allah will receive a good deed, and a good deed is multiplied tenfold. I do not say that 'Alif-Lam-Mim' is one letter, but rather 'Alif' is a letter, 'Lam' is a letter, and 'Mim' is a letter."

You are not obligated to inform the teacher about what you did in the past. If you wish to re-recite the passage from your memory, there is no harm in that. You are also not required to adhere to a specific method of recitation for memorization, as memorizing the Quran is commendable but not obligatory.