This is a pure play. By that I mean there is a precise (?) U.S. equivalent, bartender, so that the use of barman in the U.S. can be explained only by the desire to use a Britishism, or the conscious or unconscious imitation of others who have done so. (I suppose another possibility is retrograde sexism.)
According to this Google Ngram, the use of barman increased about 20 percent in the U.S. between 2000 and 2008 (actually, between 2000 and 2005; it has held steady since then):
(It’s interesting to look at an Ngram, below, showing the use of barman [blue] and bartender [red] in British English between 1920 and 2008. At least since about 1960, it appears to be a case of an encroaching Americanism, with the two terms recently nearing equality. However, barman is still used about twice as often in Britain as in the U.S., and bartender is six or seven times more prevalent in the U.S.
The New Yorker has used barman 34 times from 1937 to the present, including in a 1939 poem called “Forsaken Barman,” a 1964 Talk of the Town piece called “Barman,” and this, from a January 30, 2012, Profile by Nick Paumgarten:
He cuts off the drinks, keeps spare umbrellas on hand for sudden squalls, shuffles customers around to make space for someone’s mom, and, like any barman with a following, dispenses a lot of free drinks.
But you can find the word in all sorts of other sources as well:
“Whiskey You’re the Devil”: The best version of this traditional song is the one you sing right before the barman kicks you out. (Rosie Schaap, New York Times, March 8, 2012)
I grabbed a spot at the bar and was immediately greeted by the friendly barman, Christopher. (HoustonPress.com, March 2, 2012)