Pretty much the same way the men's quartet I sang in for sacrament a few weeks back did - straight from the hymnal. And, yes, before you ask, they are the traditional words.
For what it is worth, I grew up in a "small" town (of 100k+) outside of Los Angeles. When I was on my foreign mission and people asked me where I was from, I told them "LA" and not the name of the small town that they likely never heard of before. You see, I used a reference that they were familiar with in order to facility comprehension.
There are two major problems with this explanation for the apparent error in the BOM.
First, the Jews of North America would have expected that the Messiah would be born in Bethlehem. The migration of the Jews to North America supposedly happened around 600 BC. The Book of Micah which contained the prophesy of the Messiah's birth in Bethlehem has been dated from around 740 to 700 BC. The prophesy would have been in circulation for a 100+ years before they left. Correct me if I'm wrong but didn't they have the holy scrolls with them, so they could have just looked it up. A person claiming to be the Messiah but not correctly identifying the place of birth would have been immediately discredited. If Jesus had a bank account, do you think that the bank security questions would let him use Jerusalem and Bethlehem interchangeably as an answer to the question of his place of birth? ;-)
Second, the eight miles from Jerusalem to Bethlehem today is trivial. It was not then. The typical traveler of the day would have walked and would have taken about 5.5 to 6 hours commuting time, round-trip. Given that the automobile is the most common mode of travel today, saying Bethlehem was a suburb of Jerusalem then would be the equivalent of saying that Omaha is a suburb of Kansas City or Portland is a suburb of Seattle today. Additionally, Jerusalem and Bethlehem had separate identities and historical significance.