Apologies in advance that this answer is based on anecdotal evidence. I have not done a systematic study of either ibn Ezra or Minchas Shai, but have searched through the texts as best as possible.
In all the examples I can find, ibn Ezra discusses the gender of the noun as being different than usual, rather than assert that the "incorrect" gender of adjective (or whatever object is under consideration) is used. See, for example, ibn Ezra to Gen 2:15, Ex 35:17 (short commentary), Micah 6:9, Eccl 10:15, among others. As noted in my comment, ibn Ezra did not write the commentary on Nechemia bearing his name (it was written by Moses Kimchi, brother of the Radak), and thus the comment on Nech 3:30 asserting that שני should be read שנית, as with Esther 2:14, is not a contradiction. For what it's worth, the examples ibn Ezra point to lend credence to his assertions.
I have not seen Minchas Shai refer to a noun as sometimes masculine and sometimes feminine. He does repeat his comment from Esther 2:15 in Nechemia 3:30, but I don't see any other independent examples of this type of comment.
Minchas Shai doesn't comment on Ex 35:17, but it's possible that it was so obvious that הצר is being used both as a feminine and a masculine noun in that pasuk that it didn't bear comment. It is thus possible that he doesn't fundamentally disagree with ibn Ezra. My own thoughts are that ibn Ezra is more-or-less well supported in his assertions, and claiming that the noun is of a different gender is easier than saying that the "wrong" declension was used.
I also think that we simply don't have enough data from Minchas Shai to decide definitively what his approach was. However, I think the silence in the places where you might expect a comment (like Ex 35:17) would indicate that in general his approach is as ibn Ezra, but that in the two aforementioned places, he finds it impossible to follow this approach. I would argue that for in Nechemia, מדה שנית has been used so many times in the same chapter that it seems impossible to suggest that מדה has switched genders. Then in Esther, partially supported by the example in Nechemia, and partially by 2 Samuel 23:8 (where the adjective is changed to feminine by a קרי/כתיב, viz. פעם אחד gets a קרי of פעם אחת), he feels it more sensible to change the adjective than the noun.
As a side note: in the particular case of the implied פעם in Esther 2:14, the BDB holds that the masculine use of פעם from Judges 16:28 quoted by ibn Ezra as support for his comment is a textual corruption, along with the other instances of masculine פעם. Neither ibn Ezra nor Minchas Shai has anything to say on the other places where פעם seems to be masculine (at least, nothing to say that's survived the years).