There are a number of broadside copies at  Bodleian Library Broadside Ballads,  all printed in Dublin.  Most appear to be from the late 1860s.  There's a copy of T. Gannon's The Deserted Seamstress, printed c. 1860 by H. De Marsan of 38 & 60, Chatham Street, New York, which is set "to the tune of Lily of the West".  This would suggest that the song was reasonably well-known in America at that time, but of course it doesn't tell us which Lily of the West tune it was.

There are a number of songsheet sets of Lily of the West (all undated) at  America Singing: Nineteenth-Century Song Sheets,  most of which prescribe Caroline of Edinburgh Town as melody; for some more on the various melodies associated with that song, see  Caroline of Edinburg Town.

Although it does seem clear that Lily of the West was known in America at the time of the Civil War, the question of the tune is another matter; it may perhaps be that one of the Caroline tunes might be a more "authentic" choice than the Ponchartrain melody.

Malcolm