An option to consider would be taking US 65 south out of Springfield thru Branson and on to I-40 at Conway, AR. Then I-40 east to Brinkley, AR where you could take US 49 south to the Helena crossing of the Mississippi River, then take US 61 north to Southhaven area.
Have used that route in the past to avoid the Memphis mess and to get on to I-22 down to Alabama. Don't recall anything that would cause concern, maybe a little traffic around Branson but nothing compared to Memphis-West Memphis.
I've also been further west, at Joplin and then south on I-49 thru NWAR area to I-40 at Alma, AR, then east on I-40 to Brinkley.
Just some alternate routes.