I always fly Iberia/American/BA to Spain since American is the only airline that flies into my town. I have also found that business class tickets on Iberia (booked through the American website) are frequently cheaper, and sometimes much cheaper on Iberia than on American. American doesn’t fly out of Chicago to Madrid, only Iberia does. So my flights to Spain on American will originate in Philadelphia, Charlotte, Boston, JFK, even Dallas. That makes for three flights, but I’m used to that and it doesn’t bother me like it does some people. If you’re in business class, the seat selection problem is not really a problem, because those seats are all identical, though you may get a window when you want an interior seat and vice versa.
Out of curiosity, what is the itinerary you’re considering from New Orleans?
Have you checked the Iberia website too? Are the prices for the Iberia flight about the same on both websites? I never check the Iberia website, but it occurs to me that you might see some differences, though I don’t know.