The Rama (O.H. 292:1) writes that in Minha of Shabbat, Ashre, Uva Letzion and Va’ani Tefilati are recited. The Kaf HaHaim (ibid) notes that the Sephardic custom is to recite Va’ani Tefilati twice. This appears as well in the Siddur of Rabbi Shalom Sharabi.
When the Hechal is opened, the Siddur Tefilat HaHodesh does not mention Berich Sheme but only the Mizmor 23 (Mizmor LeDavid Hashem Ro’i) is recited.Indeed the Hida (Nitotze Orot Zohar Vayakhel) writes that Berich Sheme is to be recited only during Shaharit of Shabbat. Nevertheless, the Kaf HaHaim (ibid) explains that since Minha of Shabbat is an especially auspicious time, one should recite the special prayer of Berich Sheme at that time as well. In Morocco, it appears that both customs existed. When asked personally, Rabbi Yehoshua Maman stated that his custom was to recite it in Minha of Shabbat. On the other hand, Rabbi Baruch Toledano (Kitzur Shulhan Aruch § 270) does not mention Berich Sheme for Minha. In any event, it is not considered a deviation from a Minhag if one wishes to recite Berich Sheme as it is simply an additional communal prayer.
The Shulhan Aruch (ibid:2) continues by saying that the three verses beginning with Tzidkatecha are recited in Minha of Shabbat as well. They are recited after the Amida and are only recited on a day that, had it not been Shabbat, Tahanun would not be recited. For example, if Rosh Hodesh falls on Shabbat, Tzidkatecha is not recited. Tzidkatecha is a form of justification of Divine judgment (Tziduk Hadin) which is typically said when one passes away as an affirmation that Hashem and His decrees are perfect and just. In the case of Minha of Shabbat, Yosef, Moshe Rabbenu and David Hamelech all passed away during this time and the first, second and third verses refer to each of these people, respectively. The Tur (ibid.) writes that the Ashkenazic custom is to recite the verses Tzidkatecha Tzedek Le’Olam, Vetzidkatecha E-lohim, Tzidkatecha Keharere E-l, while the Sephardic custom is in the reverse order. He also writes that the latter custom is more correct since this is the order in which they appear in Tehilim.
The Siddur Bet Menuha, which was commonly used in Morocco, cites (pg. 230) the Seder Hayom and says that it behooves all who wish to be scrupulous in their observance to be saddened by the passing of the aforementioned righteous people. Even though Shabbat is not a time of mourning, the passing of the Tzadikim is not taken lightly by Hashem and thus, although outward acts of mourning are not observed, one should at least have in mind the passing of our righteous forefathers. One proof that there is an element of sadness in Tzidkatecha is that, as mentioned above, it is not recited on a day that Tahanun would not be recited, which tend to be happier days. Nevertheless, Rabbi Meir Mazuz (Mekor Ne’eman § 375) writes that there is no connection between the two. Rabbi Yehiel Ben Yekutiel (Tanya Rabati) writes that just as Tziduk Hadin is normally recited standing up, so too should Tzidkatecha be recited in this manner, and the Kaf HaHaim (ibid:15) concurs.
Summary: Some Moroccan communities recite Berich Sheme during Minha of Shabbat while other do not. Tzidkatecha is recited standing up and when reciting these verses, one should be mindful of the passing of Yosef, Moshe and David which took place at that time of the week.