I agree with Michael that the Blue Water Bridge in Sarnia is your best option for a crossing point. You could then use I-69 to completely bypass the Detroit area and US-23/US-223 (largely freeway quality) to I-475 around Toledo and join up with I-75 to Columbus. After that I-71 to Louisville, I-65 to Birmingham AL and I-59 to New Orleans is as straightforward a route as there is. I would not use the Natchez Trace, even though scenic, as it is a two lane road with a low speed limit, few exits and limited passing opportunities.

This is a three day drive at a minimum, with your longest day being the first one between Burlington and Montgomery. The halfway point between Montgomery and New Orleans on the routing suggested above would be roughly Decatur/Huntsville AL where there are a few campgrounds around Wheeler Lake on the Tennessee River. You should also be far enough south at that point that camping in a travel trailer is possible, depending on the actual weather conditions at the time.

Your original suggestion of I-40/I-55 to go through Memphis is almost as good. However it would move your halfway point to Jackson TN and options there for overnight trailer camping are much more limited. You could, of course, use one route southbound and another northbound to see more of the country between you and New Orleans.

AZBuck