Remember that when the Prophet Alma wrote this it was almost 600 years since any of his people had been in Israel. They have no clue of the geography of Israel at that point.  
 
Bethlehem is a suburb of Jerusalem with less than 5 miles from the center of one to the other.  
 
Would you expect Mary or Joseph to have detailed knowledge of the Americas? 
 
Of course the Americas? Do you know the name of every town in the Greater Beijing metropolitan area of China? 
 
Of course not, he used the name he knew to be closest and like the other posters have said even one as uneducated as Joseph Smith would know that Christ was born in Bethlehem not Jerusalem. 
 
If Smith was making a story up he would not of let such a simple error get through. But since he is translating a historical record from the Americas he has no ability to change what is written! 
 
I thank you for the question Josh! 
 
I hope my answer helps you!