Polygamy in the Parashah

This week's Torah portion discusses the case of a man with two wives:

Should a man have two wives – one who is loved, and one who is despised – and the firstborn is that of the despised wife… he may not grant birthright privileges to the son of the loved wife over the son of the despised wife.

The Taz made famous the concept that the rabbis lack the authority to prohibit something that the Torah explicitly permits [R. David Cohen wrote the authoritative study of this topic in his Gevul Ya'abetz, later republished in his Harhavas Gevul Ya'abetz (see here for some key points he raises)]. If so, how could Rabbenu Gershom prohibit polygamy when the Torah (above) explicitly permits it?

R. David Silverberg cites two answers to this question:

The Chatam Sofer answered by distinguishing between two types of rabbinic legislation: a gezeira (rabbinic decree), and a cherem (ban). A gezeira is the more familiar type of enactment, whereby the Jewish people’s central rabbinic authority issues laws binding on the entire nation, as they see fit in order to ensure proper observance of the Torah (such as refraining from blowing the shofar when Rosh Hashanah occurs on Shabbat). A cherem, however, such as that imposed by Rabbenu Gershom, operates much differently, on a communal, rather than national, scale. Citing from the Ramban’s Mishpat Ha-cherem, the Chatam Sofer explains that a cherem is essentially a communal neder – a provision taken on by a community, corresponding to a personal oath taken by an individual. Rabbenu Gershom’s cherem was exceptional in that its sphere of influence covered all of German Jewry, rather than a single community. Fundamentally, however, it operates according to the same halakhic mechanism as a standard cherem, a provision taken on by a community at the behest of its rabbinic leadership.

Therefore, the explicit reference in the Torah to polygamous marriages did not stand in the way of Rabbenu Gershom’s ban. Just as an individual can take a personal neder to refrain from something explicitly permitted by the Torah, so can a community accept a cherem banning an activity which the Torah expressly permits.

Another answer, that denies that there is even a question, is:

A simpler answer is suggested by Rav Yosef Schwartz of Grosvarden, in his Ginzei Yosef (154), who points out that the Torah does not, in fact, explicitly sanction polygamy in this verse. The Torah here does not specify when the man married the second wife; it perhaps speaks of a case where one married the more beloved wife after having divorced the despised wife or after her death. The verse can just as easily be read to mean that the husband married the two wives successively, and thus it does not make any clear reference to a polygamous marriage.

I'll b"n check in Harhavas Gevul Ya'abetz to see if he has anything to add to these answers.