"The Book of Mormon is made up because Joseph Smith wrote that Jesus was born in Jerusalem."
The early books in the Book of Mormon talk of "the land of Jerusalem". Likewise, I lived in a suburb of Cincinnati for years, but we often told people we lived in Cincinnati. I was raised 20 miles south of Provo and 60 miles south of SLC, but I often told people in college that was born and raised "in the Salt Lake City area" - and I often tell people I lived in Boston while I attended college there, even though I lived in Medford, Somerville and Woburn (and actually attended college in Cambridge). If someone is speaking to a group of people who have never been to the region being described, it is totally natural for them to use a nearby, well-known city as an approximate substitute.
Joseph Smith knew that Jesus was supposed to have been born in Bethlehem, so there is no logical reason for him to have written Jerusalem if it meant within the actual city limits.
This is one case where I have absolutely no problem whatsoever with the passage in question, and I actually believe it bolsters authenticity more than it decreases authenticity (or, at least, is completely neutral). In other words, I would expect a record like the Book of Mormon reference Jesus' birth being "the land of Jerusalem" in that way.