As always, please consult your rabbi for a practical ruling.
There are several issues here.
- Does "Leining" count as "singing?"
- Is the fact that she's reciting Tanach make a difference?
- Is there an issue of Kavod haTzibbur?
- Are there any other mitigating factors?
The Rama (O.C. 75:3) and Bait Shmuel (21:4) state that this prohibition applies only to a woman’s singing voice and not to her speaking voice.
So if Leining is more like speaking than singing, it wouldn't be a problem.
The Seridei Eish (2:8) defends the practice of women singing zemirot on Shabbat by citing the Sdei Chemed (Klalim, Maarechet Hakuf, 42) who quotes the Divrei Cheifetz who asserts that the Kol Isha prohibition does not apply to women singing Zemirot, because men do not derive pleasure from the woman’s voice in this context. In fact, the Pasuk (Shoftim 5:1) records that Devora the prophetess sang a song of praise to Hashem together with Barak the son of Avinoam.
The Gemara (Megila 23a) states that women are forbidden to receive an Aliyah to the Torah because of Kavod Hatzibbur (and in the days of Chazal the one who got an aliya would lein the Parsha). Since it does not mention Kol Isha as the reason to forbid women’s Aliyot
it is possible to argue that the Kol Isha restriction does not apply when a woman recites sacred texts.
Rav Mosheh Lichtenstein argues that the ruling of the Divrei Cheifetz and Seridei Eish permits listening to a woman singing at official Tzahal events that are formal and serious. He first bolsters the opinion of the Divrei Cheifetz and the Seridei Eish by demonstrating that Rambam, Rashba and Ra’avyah agree that the Kol Ishah prohibition is not absolute but rather emerges from concern lest it lead to sin. Thus, in a context that concern for sin is not relevant, the Kol Ishah does not apply.
It may be possible to extend R' Lichtenstein's reasoning to other formal and serious events, such as the one in the question, especially given that recitations of Tanach should not lead to concern for sin.
According to Rashi women may read the megillah for men. (Source TBA)
This appears to be a clear parallel. Apparently other Rishonim in Masechet Megillah also say that leining isn't kol isha, but I haven't checked.
3) Kavod haTzibbur is complicated. Some say it's subjective, and the Tzibbur can be mochel \ If they don't mind it's not an issue; Others say it's objective, and it is what the seforim\poskim say it is, so if the poksim say this is bad, then it is, even if the Tzibbur doesn't care.
Rav Moshe Feinstein (Teshuvot Igrot Moshe O.C.1:26) and Rav Ovadia Yosef (Taharat Habayit 2:270) rule (based on the Mishna Berura 75:17) that in case of need, one may rely on the ruling that the prohibition of Kol Isha does not apply to girls who are not Niddot. Rav Moshe writes (in 1947) that one may assume that there is no question with girls below the age of eleven. Rav Moshe writes that men must be strict regarding girls older than the age of eleven, since there are girls who “nowadays” become Niddot at the age of eleven.
It is possible that a girl who is just turning 12 is not yet a Nidda (although the only way to know for sure is to ask her).
It's mentioned that Rav Shlomo Zalman Auerbach, Rav Eliezer Waldenberg, and Rav Yosef Shalom Eliashiv cited in Nishmat Avraham 5:76-77 address this issue, but I don't have access to that sefer.
Quotes source: http://www.tabc.org/kol-torah/article/index.aspx?pageaction=ViewSinglePublic&LinkID=712&ModuleID=43&StartDate=4/5/2013&NEWSPID=1 (alternate location: http://koltorah.org/ravj/The%20Parameters%20of%20Kol%20Isha.htm )
Women chanting the Torah or Megilla with the appropriate notes (ta’amei ha-mikra) is not included in the prohibition of kol be-isha erva.
See: R. Jacob Hayyim Sofer, Kaf haHayyim, sec. 689, no. 2; Resp. Divrei Heifets, cited by Sdei Hemed, Klalim, Ma’arekhet kuf, klal 42; R. Jehiel Jacob Weinberg, Resp. Seridei Eish, II, sec. 8; R. Nahum Tsvi Kornmehl, Resp. Tiferet Tsvi, II, sec. 7; R. Samuel haLevi Wosner, Resp. Shevet haLevi, III, sec. 14 – who indicates that most rishonim are lenient by keriah de-mitsvah; R. Ovadiah Yosef, Yehave Da’at, III, sec. 51, note, and IV, sec. 15, end of note; R. Ovadiah Yosef, Resp. Yabia Omer, VIII, O.H., sec. 22, no. 10 and IX, O.H., sec. 98, no. 9, and sec. 108, no. 74; R. Ovadiah Yosef, Me’or Yisrael, I, Megilla 4a, s.v. “beTosfot d”h Nashim,” p. 251, and Megilla 23a, s.v. “Tanu Rabbanan, haKol,” p. 279; R. Ovadiah Yosef, Halikhot Olam, II,
Ekev, sec. 2, note 2, p. 74; R. Ovadiah Yosef, MeShiurei Maran haRishon leTsiyyon,
Note: I have not validated nor confirmed any of the sources in the above paragraph.