The sefer Harirai Kedem suggests a different understanding of the machlokes. Everyone, the author of this sefer writes, agrees that Tisha B’Av is always on the ninth day. That’s why the Rema maintains we observe aveilus betzina. As for the Mechaber, he evidently maintains that there can only be one day on which the aveilus of Tisha B’Av should be observed. And since we cannot fully observe the aveilus on Shabbos, we observe it on Sunday.
As for the halachos of shevua she’chal bo: The Mechaber apparently maintains that the halachos applying to the week of Tisha B’Av are connected to the aveilus of Tisha B’Av, not to Tisha B’Av itself. The aveilus of the week that Tisha B’Av falls out in is a prelude to the aveilus of Tisha B’Av. So even though the actual day of the fast is on Shabbos, since aveilus is not observed on Shabbos, its week also does not have any halachos of aveilus.