Cohan's "The Yankee Doodle Boy" is from 1904. "Mary's a Grand Old Name" is 1905. Both would fit, I think.
Other possibles: When You Were Sweet Sixteen, 1898 A Bird in a Gilded Cage, 1900 In the Good Old Summertime, 1902 Meet Me in St. Louis, Louis, 1904 In My Merry Oldsmobile, 1905 Wait Till the Sun Shines Nellie, 1905 Harrigan, 1907 Has Anybody Here Seen Kelly?, 1909 Let Me Call You Sweetheart, 1910 Macushla, 1910 Alexander's Ragtime Band, 1911 I Want a Girl (Just Like the Girl that Married Dear Old Dad), 1911 When You Wore a Tulip (And I Wore a Big Red Rose), 1914 Oh, Johnny! Oh, Johnny!, 1917 For Me and My Gal, 1917 Last, but not least, perhaps the most popular song of the era (at least in the soldier context), but at the end of the war:
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