There were times when the Rebbe agreed with arranging a taanis shaos and even said a sichah on some of them. However, when the Yom Kippur War took place, he strongly held that Tishrei is not the time to fast, citing the Tur and Beis Yosef to this end.
The following article is an interesting background report by Rabbi Avraham Shmuel Levin, the secretary of Agudas Harabbonim of America for many years. He relates how some Rabbonim attempted to establish a fast day to be held at the end of Tishrei and how this was circumvented in a Lechatchilah Ariber’dike way.