Sanhedrin 51: The Bad End of the Kohen's Daughter

The death sentence for one who desecrates Shabbat (intentionally, with witnesses and warning) - stoning or burning? (the conclusion is stoning, but that's not R. Shimon's opinion). But what if the Shabbat desecrator is the married or betrothed adulterous kohen daughter? Is her execution stoning or burning? Would a kohen who serves in the Temple have greater leniency with regard to Shabbat? What about a case of incest? Also, Rabbi Yishmael eases up a bit, at least with regard to the accoutrements of investigation. For example, when the kohen's daughter's behavior turns out to be less serious than these egregious affairs. Plus, the conspiring witnesses who try to bring her down. But why is the kohen's daughter the focus here?!