R. Avrohom Bornsztain has a responsum (Avnei Nezer O.C. #509) discussing women's obligation in this commandment. He wonders why the Sefer Hachinuch (cited in the question here) states that women are exempt because they are not war-wagers yet in Mitzvah # 425 (or 423) he states that women are obligated in the destruction of the seven nations.
R. Bornsztain then makes a distinction between the nature of the commandment for men and women. The Sefer Hachinuch mentions that the obligation of eradicating Amalek is incumbent upon every individual. That is why it would apply to women – even though they don't go to war, they can still kill an individual person if they see one. However, if it's just an individual killing and not an act of war, then it would be no different from any other death penalty which is not administered on Shabbat. Since (for women) the commandment is not operable on Shabbat, it is considered a time-bound positive commandment from which women are exempt.
Eradicating the seven nations, on the other hand, is for the purpose of preventing them from having a negative influence on us. As such, killing them would be a melacha she'aino tzericha legufa and therefore biblically permitted on Shabbat. Since it is then not a time-bound commandment it is incumbent upon women as well.
What emerges from this is that eradicating Amalek would not supersede Shabbat when done as an individual killing, just like all other death penalties, but it would supersede Shabbat when it was part of a war.
