This is great timing for this question, as I have been wondering the same for the last couple days. Different location, same issue. To get from New Orleans to St. Louis by Amtrak involves a bus ride from Carbondale IL, and there is no baggage service on the bus. Its about 140 some miles to St Louis, was thinking about just riding it, providing I could get my bike off at the Carbondale station. No one I have talked to on the Phone seems to know. Better to ask here I thought than open a new thread.
I do find it interesting that you can get to the St Louis station from New Orleans if you are willing to go to Chicago first. Thought about doing that, but a 25 hour train trip sounds like more hassle than an extra 140+/- mile ride, except that it will be on crummy roads.