The second point raised by the Turei Even, that bikurim are zman gerama because they are only brought between Shavuos and Chanukah, is easier to answer. Zman gerama is a limitation on when a chiyuv applies. By bikurim, the chiyuv applies 24x7. However, to fulfill that obligation requires a certain type of good fruit, i.e. a proper cheftza shel mitzvah. The time span of Shavuos to Chanukah is when such fruit is available, a demarcation of the cheftza shel mitzvah, not a limitation on the chovas hagavra.
The Sefer haChinuch interestingly writes that women are exempt from the mitzvah of bikurim. One could argue that the Chinuch’s position is motivated by either of the two questions of the Turei Evev. However, this position is still difficult to understand. Recall that the Mishna only said women are exempt from reading the parsha of viduy bikurim, implying, that they are obligated in the mitzvah of bringing bikurim. Even if one argues that this obligation is only Rabbinic, it begs the question of why the Rabbis would obligate women in this mitzvas aseh she’hazman gerama and no other (Minchas Chinuch).