Wow - so interesting!
I wonder if the witnesses on the marriage certificate might hold a clue to the identity of your Friedrich Gottschalk. Is Miss Charlotte Leininger a sister to Anna CHRISTINE Leininger? The men could be brothers-in-law or cousins or friends from his point of origin in Prussia. A long shot, I know - but they are worth listing if you do post a query in another message board (such as Germany board). Do you have a date of immigration from any source (such as a family story)? How do you know he lived in New Orleans first - family story? I guess your best bet at getting help (and not from me, sorry to say, I'm stumped) will be to share all that you know about your Friedrich and work back from there.
Witnesses on the marriage license:
Robert Edgle (?)
Thanks for sharing this, and best wishes.