The Mississippi is, of course, navigable for most of its length, and as the US Navy showed in the American Civil War, oceangoing ships of the nineteenth century had no trouble going up.
The issue is whether New Orleans could be fortified so as to deny navigation. The Confederates were none too successful at that IOTL, but they only had a year to do so.
I wonder if you couldn't circumvent it by digging a canal. Suppose a canal were dug from Vicksburg to Jackson and then down the Pearl River--would it enable navigation from the Mississippi River to the Gulf through the state of Mississippi? It would certainly be a large undertaking--but then, so was the Erie Canal at this time. Could a canal fit for ocean-going ships be built, or would it be limited to river barges?
Outright conquering New Orleans would be cheaper, I think, unless this is a world where Napoleon is triumphant and France can dictate terms to everyone.