Did Muhammad attend a Church?

As I'm taking a class on Classical Islam this semester, I'm thinking more and more about the dialogue between religions that went on in the Patristic era. Muhammad certainly had interactions with Christians, but on what level?

Islamic tradition says that Muhammad knew an Assyrian Monk named Bahira. He is later identified by Christian authors (St. John Damascus, for instance) as an Arian. As far as Muhammad's Christology goes, this isn't unthinkable.

But beyond just knowing the monk - is there any indication that Muhammad had delved further into Christianity? I think the answer is yes. Muhammad is noted as being "unlettered" in the Qu'ran and in Muslim tradition. Whether this means that he was illiterate or just knew basic writing isn't my concern. My concern is where Muhammad got the stories from Surahs 3:49 and 5:110. I've noted in a previous post that both stories occur in "Gnostic" (for lack of a better word) texts - the Infancy Gospel of Thomas and the Gospel of the Infancy of Jesus Christ.

How would Muhammad, an "unlettered" man have come across these texts? I'm proposing that Muhammad would've heard them - in the liturgy. Granted, one can imagine Bahira discussing these texts with Muhammad, but why would Muhammad include the stories if there weren't some attachment to them?

This is really speculative, but I think it's at least a theory that could be explored.