Since this posting I found 1850 and 1860 census records in Louisiana (Shreveport and Mansfield, respectively)
from them the year of departure from Ireland to Louisiana would have to be from 1844 to 1850 as the 1850 census records has Jemima Constantine as 6 years older and the daughter of James and Margaret Constantine (all three born in Ireland).
James Constantine's birth year would be either 1816 or 1818.
He was in Dublin in 1843.
Still looking to find his parents somewhere in British Isles.