While I've not done the drive completely from point-to-point that you're asking about, I've done a goodly part of it. I would do a combo of both. And of course, there are alternatives to the interstates,as well as alternative interstates, but since you asked specifically, I figured I'd answer specifically.
I-70 to St Louis (I would actually take I-64 to I-270 to bypass St Louis), I-55 south to I-155 to US 412 to I-40 to I-26. Why? To completely avoid Arkansas and in particular going through Memphis. I've done that drive many times from Little Rock to Memphis, and Memphis has always been frustrating, regardless of the time of day.
I've done the drive to Greenville SC and it's not a bad drive, as interstates go. There's some alternatives to use to bypass some of the bigger cities (KC, Nashville, Knoxville). My guess is, this is not a sightseeing drive, you want to get there rather quickly. This would do it.
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