Even though the Yerushalmi tells us that it is prohibited to mourn on Shabbos, the mourning over Moshe is different like the gemara says that the mourning for Moshe was especially intense. This is showcased in certain foods we eat on Shabbos. The custom of eating eggs, which is a mourner’s food, on Shabbos is because Moshe passed away on this holy day. This custom is also cited in Shyarei Knesses Hagedola who writes that it is even better to eat the eggs by the third meal of Shabbos, when Moshe passed away.
R' Rafael Meizlish writes that the reason we eat kitniyos, beans (referring to the beans of cholent) on Shabbos (which is a mourner’s food) is due to mourning for Moshe who passed away on Shabbos.
Tosafos cites Rav Sar Shalom Gaon that the reason we say tzidkascha (“Your righteousness”) on Shabbos at Mincha is because that is when Moshe passed away. The Bach adds that the Zohar informs us that Yosef and Dovid Hamelech also passed away on Shabbos at Mincha time. As a result, we say [pesukim that begin with the word] tzidkascha three times.
