Worse, from a strategic point of view, Mustasim had recently angered the Shiites by various insults and offenses, such as throwing the poem of a famous Shiite poet in the river. Now vengeful Shiites volunteered help to the Mongols in Mosul and other places along their march. The caliph’s vizier, or chief minister, was himself a Shiite of uncertain loyalty. Islamic opinion afterward held that the vizier, al-Alkamzi, vilely betrayed the caliph and conspired with the Mongols; an exhortation in Muslim school books used to say, “Let him be cursed of God who curses not al-Alkamzi.” As fighting began, Hulagu, acknowledging the importance of Shiite support, prudently posted guard detachments of a hundred Mongol horsemen at the most sacred Shiite shrines in Najef and Karbala.
I'm doing some searches on JSTOR but I can't find any primary sources in translation to back it up. Someone who knows the histories a little better could do a better job finding one.
I know al-Alkamzi is given various motives for his advice - which was not to raise a huge army to try to to defend the city, nor to negotiate with Hulagu (who it should be mentioned was riding with a truly massive army, not just an ordinary Mongol blitzkrieg) - and he's particularly vilified.
That being said, recruiting locals was a favorite Mongol tactic. Hulagu had previously offered the standard Mongol peace - pay taxes or die horribly - and the caliph hadn't even deigned to discuss terms with him, instead threatening and dismissing the most powerful military commander on the planet.
I don't remember the Secret History mentioning these Shi'a, which isn't surprising. However, I find it unlikely that this split was wholly sectarian - given the disinterest of the Empire in Islam, or really any of the religious groups they conquered. I would not be remotely surprised if these joiners were really a varied group of local recruits that were used by Sunnis to vilify Shi'a, particularly because references to Alkamzi often have anti-Shi'a subtexts. The Mongols are extremely hated in period Muslim texts, partly because of their brutality, but more because they were infidels who represented an existential crisis for Islam, which was conceptualized as spreading until it was the only religion on Earth.