Sanhedrin 96b says that descendants of Haman taught torah in Bnai Brak. Can Amalek convert??

The apparent contradiction is asked by many Torah sources, among them the Iyyun Yaakov and Rav S. Alkabetz in his Manot HaLevi on Esther: On one hand, the Talmud (Sanhedrin 96b, Gittin 57b) states that the descendants of Haman taught Torah in Bnai Brak, and yet the Mechilta (end of Parshat Beshalach) tells us that “G-d swears that we do not accept converts from Amalek.” The Rabbis also discuss this question based on the opinion of the Rambam (Laws of Kings 6:1-4) that seems to be that we do accept Amalek converts. Some explain the Rambam’s view that indeed he goes against the Mechilta, understanding the above Talmudic passages as countering the Mechilta (the Chida in his Ein Zocher). The Avnei Nezer (Orach Chaim II:508), Or Sameach (in his Meshech Chochmah, end of Parshat Ki Tetze)) and Tzitz Eliezer (XIII:71) explain that the Rambam himself posits that Amalek can only become a Ger Toshav (resident stranger in the Land of Israel) and not a ger tzedek (convert). The Netziv of Volozhin (on the Mechilta) and the Hazon Ish (on the above Rambam) posit that there is no contradiction between the Talmud and Mechilta, since the Mechilta is referring to a time of war – only then do we not accept Amalekites. There are others, however, that explain the opposite, i.e., that both the Mechilta and the Talmud hold that Amalek cannot convert, and various explanations are given as to how the descendants of Haman taught Torah in Bnei Brak (Rav Hayyim Palaggi, Rav Y. Engel and the Hazon Ish – cited in Tzitz Eliezer above). One such solution is that indeed Amalek cannot convert, but his grandson can. How? A female Amalekite who marries a non-Amalekite non-Jew and gives birth to a son, that son is not considered an Amalekite. Thereafter these descendants can convert to Judaism (Rav Engel and MaHarSham.III:272). Rav Yehoshua of Kotno (Yeshuot Malko, Likkutei Shu”t 15) states that after the fact (if an Amalekite converted) the conversion is accepted. The Tzitz Eliezer concludes that “it is especially important to notice the words of the Baal HaTurim (Shmot 28:7) from which it is clear that he had the text in the Talmud, not ‘the descendants of Haman’ rather ‘the descendants of Naaman’, as already pointed out by Rav Margliyot in his Mekor Hessed commentary to Sefer Hasidim, sec. 1019, and thus there is no proof from our Talmud against the Mechilta…”