Reciting Shema after its time for a mourner during Shiva

Is it permissible for a mourner during Shiva to recite Shema after the time of Shema according to the Gaon?

Even though after the time of Shema, one who recites it receives a reward like a person reading the Torah, nevertheless, one must recite Shema in their prayer.

In the Gemara Berachot, page 9b: "Rabbi Yehoshua says: until three hours, as it is the way of kings to rise at three hours. One who reads from then on does not lose, like a person reading the Torah." The opinion of the Kesef Mishneh (Chapter 1, Laws of Shema, Law 13) is that the time for reciting Shema from the Torah is all day, and by rabbinic decree, the reward for reciting Shema is specifically within three hours, and afterwards, the reward is like reading the Torah. However, in the Shulchan Aruch, it is ruled until four hours, and afterwards, the reward is like reading the Torah. And the Mishnah Berurah writes in Siman 71, Se'if Katan 4: "If the state of mourning continues until close to the end of the time for Shema, which is until a quarter of the day according to the hours of the day, and if he prays in order, the time for Shema will pass, let him recite Shema without blessings, and then pray in order and recite Shema again with the blessings." And even though he later recites Shema even if he already recited it, it seems that the method of the Mishnah Berurah is to recite Shema, as written in Mishnah Berurah, Siman 60, Se'if Katan 4, that even if he recited Shema before the prayer, let him recite it again in his prayer, to stand in prayer from words of Torah. And even though it is forbidden to engage in words of Torah, it is considered part of the prayer. And therefore, after the time of Shema, let him recite Shema to stand in prayer from words of Torah, and even though it is forbidden by Torah law, nevertheless, it is part of the prayer. And what the Mishnah Berurah wrote in Siman 599, Se'if Katan 5, that in the Mincha of Tisha B'Av, one does not recite Shema because it is forbidden to study Torah, this is because it is not part of the order of prayer, it is forbidden.