I would like to turn around these two questions. We saw that it is problematic to ask personal requests on Shabbat, and certain verses of Avinu Malkeinu correspond to the blessings of the weekday Amidah. However, our Mizrachi brothers do say it on Shabbat. What is their reasoning, what sources do they have?
The Ra'van on Yuma writes: ( ראב"ן יומא ד״ה ובמוצאי של יוה"כ )
וגם אומרים אבינו מלכינו לערב שבת ושבת שבין ר"ה ליוהכ"פ
And we say Avinu Malkeinu on Friday and on Shabbos between Rosh Hashana and Yom Kippur
(Indeed while the Rama (OC 602) sounds like Avinu Malkeinu is said on Friday afternoon before Shabbos Shuva, the Yad Ephraim in 554 says the custom in his locale was not to. In any event it seems some Eidos Mizrach communities do say it even on Shabbos Shuva itself.)
I haven't seen a reason for allowing this. Perhaps the reasoning of the Baal HaTanya can be applied here, since Shabbos Shuva is an Eis Ratzon, and if not when?
In the Eidus Mizrachi Siddur I have with the Haskama of HaRav HaGaon Ovadia Yosef ztz"l they mention saying Avinu Malkeinu on Shabbos Shuva, they omit three of the lines (they're written smaller).