Cutting Nails During the Thirty Days of Mourning for the Purpose of a Mitzvah Immersion

A woman whose immersion night falls during the thirty days of mourning for her brother, how should she handle cutting her nails?

Beit Yosef, Siman 300: "The Maimonides glosses, Chapter 5 (DK 5:2) in the name of R"M of England: A woman whose immersion occurs on a festival should not cut her nails herself, but may instruct a non-Jew to cut her nails, even with scissors, since instructing a non-Jew is a rabbinic prohibition, but in the case of a mitzvah, it is not prohibited. However, she cannot cut them well with her teeth or hands. The same rule applies if the mitzvah immersion occurs within thirty days after the seven days of mourning."
The Shulchan Aruch also ruled, Siman 300, Seif 7: "A woman whose immersion occurs after seven days, within thirty days, if she cuts her nails with her hands or teeth, she does not cut them well, but should instruct a non-Jew to cut them with a razor or scissors." The Rema added: "Not only a non-Jew, but the same applies to a Jewish woman, as during the festival all are prohibited from work, hence it is said specifically to a non-Jew, but in mourning, she may instruct a Jewish woman."
In Taz, there, SK 3, it is written that through a Jewish woman it is permitted due to change, like with hands or teeth, which is allowed for all in mourning. However, he questions why the Rema wrote in Siman 303 that it is permitted to cut nails for circumcision and for any mitzvah, and why it is prohibited here by herself? And concludes: "We have not seen women act this way to cut nails through another within thirty days."
In Nekudat HaKesef, it is written that through another it is not considered a change, and to the question of Taz, that immersion is considered a mitzvah, and why it is prohibited by herself, he answered that whatever can be corrected is preferable.
In Shach, there, SK 4, it is written to disagree with the Rema that it is prohibited through a Jew, as he is the mourner's agent, and only through a non-Jew is it permitted, since instructing a non-Jew is not prohibited except rabbinically, and in the case of a mitzvah, it is not prohibited. In practice, if possible through a non-Jew, it is certainly preferable to follow the opinion of Shach and Shulchan Aruch. If not possible through a non-Jew, it is preferable through another Jewish woman, and if not possible through another, she may be lenient even by herself, as written in Aruch HaShulchan, Yoreh De'ah, Siman 300, Seif 6: "But there is an opinion that the main thing is that even through a Jewish woman it is permitted, and even by herself it is permitted to cut if she cuts them better, and it seems that this is the custom." It is also written in Chochmat Adam, Klal 165, Ot 30: "If it is not possible through another, it seems to me that it is permitted to cut by herself (as there are those who say that a married woman is also permitted to iron and cut, see Beit Yosef, Siman 381, regarding makeup and adornment, and in Siman 300)."