Is it permissible to say: "O Allah, have mercy on us through the intercession of Your Prophet Muhammad - may Allah's peace and blessings be upon him -"?

There is no harm in this supplication because the intercession of the Prophet (peace and blessings be upon him) for his Ummah is established by the Sunnah. Requesting intercession should only be from Allah the Almighty, by saying: "O Allah, make Your Prophet intercede for us," and not by saying: "O Messenger of Allah, intercede for us." For this has not been mentioned in any book, Sunnah, or practice of the Salaf.