I hope to see myself in a situation similar to yours next year!
My understanding is that the student population at LSUS is quite a bit more homogeneous than the population at Tulane. With that said, I've noticed that match lists for Tulane tend to be more geographically spread out than the lists for either of the LSUs. I believe according to Wikipedia, Tulane also produces somewhere around 1/3 of all MD/MPH graduates. I would think this makes the school known, if not necessarily in the medical community, in the transitional community between medicine and public health. I would find this to be a bit important in terms of potentially moving up an administrative ladder or something.
With that said, you say that cost isn't an issue, but Tulane is the second most expensive medical school in the nation (right behind Tufts!). LSUS is a huge bargain in comparison. Personally, I see LSU-NO as a great compromise between cost and recognition. LSU-NO, in my very limited looks at match lists, tends to have more geographic diversity than LSUS.
Best of luck in your decision, and I look forward to following your choice!
PS...Louisiana medical students have a lot of blogs out there...and I mean pretty famous ones. PandabearMD went to LSUS. Tulane has a few blogs too, but the writers that I know of these blogs don't directly state that they go to Tulane, so I'm not going to list them here.