I would be willing to hazard a guess that Sidney Rigdon-—an educated but heretical Baptist minister whom I believe to be the true author of the Book of Mormon-—was quite knowledgeable of Islam.
That period of US history came hot on the heals of the Barbary Pirates conflict which got many Americans to study Mohammedism in an effort to understand the enemy. Even Thomas Jefferson owned a translated Koran. Rigdon was an intellectually curious man; an English translation of the Koran sounds like something that would have been right up his alley.
People like Rigdon and his protege` Joe Smith were interested in establishing an authoritarian system with a religious facade-—a system mainly about power, sex, and material gain. What better model to follow in pursuit of those aims than Islam?
Unlikely, since the book was published before SR came into contact with it.