Why doesn't Surat At-Tawbah begin with the Basmala?

Because the Basmala is a covenant of peace, and Surah Bara'ah was revealed with the sword. It is also said that it was the custom of the Arabs that if there was a treaty between them and a people, and they wished to annul it, they would write a letter without inscribing the Basmala therein. When Bara'ah was revealed, annulling the treaty between the Prophet and the polytheists, the Prophet sent Ali ibn Abi Talib with it, and he recited it to them without the Basmala. It is narrated from Uthman that Al-Anfal was among the first surahs revealed in Medina, and Bara'ah was among the last to be revealed in the Qur'an. Its story was similar to that of Al-Anfal, so it was thought to be part of it. The Messenger of Allah (peace be upon him) did not clarify that it was part of it, so they were joined together without writing the line of the Basmala.