The Cult of Mithras was an exclusionary secret society more akin to the Freemasons today. Everyone knew they were there, but you had to be invited in. It does sound cool, with its masks, robes, hierarchy of membership, secret rings and handshakes, but these are all designed to keep people out. Unless there there is a huge shift in its organisation it will likely never be willingly spread to the general populace by its adherents. Christianity's main rivals in the Roman empire were the Cult of Isis and Manicheanism a bit later.
A Western empire dominated by a Cult of Mithras likely already happened. It was a cult for the military elite and other nobles so with a string of barracks emperors or other military commanders calling the shots at the top of the empire they would meet up in their secret temples and ensure they maintained cushy positions. Perhaps there were people on soap boxes of the time decrying the empire was secretly being run by a mysterious cult of a foreign god and they would be right! The cult of Mithras had its time and did little with it.
Due to her focus on the poor and protector of sailors the Cult of Isis ensured most ports had a temple or shrine to her and early Christianity depicted Jesus as a more motherly figure to rival the Isis Cult and other matriarchal cults in the region. This stopped after the whole Madonna imagery gained prominence, cribbing directly from Isis' imagery of her and her son Horus.
Manicheanism, a syncreticised Zoroastrianism with Christianity and some unique touches had a strong following in the Western Empire that was stamped out due to some conversions of the likes of Augustine of Hippo. He wrote on how to identify followers of the prophet Mani due to their "Manic" behaviour (a term from these virtually forgotten heresy hunts).
