Something seems screwy.
He didn't escape to the U. S. until 1869, and you are right, his "Southern Seas" poems appeared in 1873. This, of course, was a book; he could have submitted poems to the literary magazines which were very popular.
I have not seen his biography by Evans, "Fanatic Heart."

It seems impossible that the poem appeared in 1862, if it indeed was by O'Reilly. We need that article by Warren in Civil War History, vol. 47, where the author tells when the poem appeared. The date also could be in the Cambridge poetry volumes; I believe that there are reference sections to the poems.

Press on!