Is memorizing Surat Al-Baqarah or Al-Imran, or performing Hajj, a prerequisite for marriage approval, given the fear that such memorization might not be solely for the sake of Allah?

If you desire to marry her for her religiosity, and you agree to her condition, seeking the countenance of Allah and the completion of a good deed, and that you marry one who helps you in obedience and whom you help in it, then there is no harm upon you. This is striving for the Hereafter, as Allah the Almighty said: "But whoever desires the Hereafter and strives for it with the appropriate striving while he is a believer – then [for] those, their striving will be appreciated." (Al-Isra: 19). And Abu Talhah married Umm Sulaim, and her dowry was his acceptance of Islam. Strive to memorize the Holy Quran in obedience to Allah, and there is no impediment to the sister's request being a motivator for you, as this does not contradict sincerity. It is similar to one who enters religious competitions for that to be a motivation for memorizing the Quran. The Permanent Committee has permitted prizes for religious competitions to encourage knowledge and memorization of the Quran, and the believer should be sincere to Allah.