I located the succession papers (as they call probate in Louisiana) for John F Crane and Alice Whelan Crane; the documents were not filed until after Alice's death in 1903. Both died intestate.
According to the file, John's middle name is given as Frederick, so I do believe this is the one I'm looking for; he died 11 Sep 1875.
There's also an inventory of property owned by the Cranes, six lots in all. The earliest was purchased by John in 1853, so that gives us some indication of when he got to New Orleans, and suggests that he probably wasn't there in 1850.
I did find a John Crane in the 1850 census in New York, listed as born in Ireland, working as a bartender. No other apparent connection to the John Frederick I'm looking for, but I can't discount it completely yet.
I did note that Alice was very definitely Catholic, and John F was Anglican (if I'm correct about St George's parish), for what that's worth.