The origins and history of the current practice, to repeat the verse and read זכר with a double segol, are summarized in a 1998 article by Yosef Peretz on a Bar Ilan website, based on the work of M. Breuer and Y. Penkower.
The story is briefly as follows:
The Radak was the first to testify to seeing both zaykher ("five dots") zekher ("six dots") in various biblical manuscripts.
The Maaseh Rav (134) of R. Issachar Baer of Vilna, the earliest anthology of the Vilna Gaon's minhagim, claims that the GRA pronounced the word זכר as zekher when he recited the verse on Shabbat Zakhor. However, R. Issachar Baer immediately adds the testimony of R. Hayyim Volozhiner -- the GRA's close disciple -- that the GRA in fact pronounced the word as zaykher, the same vocalization found in all standard chumashim; i.e., five dots, not six (R. Hayyim repeats this claim in a haskamah for Maaseh Rav dated 1817). The Mishnah Berurah recommends reading both pronunciations.
However, the oldest and most reliable versions of the Masoretic text of the Bible all read zaykher -- with five dots.
The custom of repeating זֵכֶר עֲמָלֵק only became widespread in the late twentieth century. It seems to me that out of respect for the Masoretic text -- in the entire yearly Torah reading, this is the only instance of a repeated variant pronunciation -- and to avoid what is most likely a מנהג טעות, there is far more reason not to repeat the verse than there is to repeat it.
Closely related to this discussion is ArtScroll's usage of zekher in the Ashrei prayer, in every edition of their Siddur (the Chabad Siddur also uses this pronunciation). This too is based on a purported custom of the GRA, as recorded in Maaseh Rav (28), and here too R. Hayyim Volozhiner refutes the testimony of R. Issachar Baer.
There is no textual basis for this vocalization, either in the Masoretic Bible or in any variant manuscript, all of which have זֵכֶר in Psalms 145:7. Like the zaykher/zekher repetition of Parashat Zakhor, the ArtScroll Siddur imposes an obscure practice on mainstream traditional Judaism, without due reverence for its centuries-old liturgical tradition.
Postscript - אחר כך מצאתי:
See also R. Menachem M. Poliakoff, Minhagei Lita - Customs of Lithuanian Jewry (Baltimore: 2009), p. 53:
Despite the suggestion of the Mishnah Berurah to repeat the word zeicher to accommodate the opinion that the reading is zecher, in the Telshe Yeshivah and most other places in Lithuania they did not repeat it. In fact, according to former talmidim, they did not observe this custom even in Radin, where the Chafetz Chaim (author of the Mishnah Berurah -dsz) had his Yeshivah.
My humble opinion is even if this repetition is permitted, it is certainly bad practice because it publicly introduces an element of doubt into the accuracy of the mesorah.