In addition to Wolfhaley's suggestions:
Cedar Hill Park in Nashville: Enjoyable and consistently challenging round.
Inverness Disc Golf Park in Birminhgam: Well-designed course on a beautiful plot of forested land with plenty of elevation. This was right in my wheelhouse.
Courses on your direct route that I have not played but hope to:
Severen Lang, Charlie Vettiner, and Ryan's Course around Louisville KY
Holler in the Hills (DGCR #24, but check current conditions) near Brownsville KY
Oggwood in Cedar Hill TN
Cane Ridge x 2 in Nashville TN
Kudzu Cove, east of Huntsville AL
Clay DGC near Birmingham AL
You could also contact Wellsbranch250 about playing V-Rock in Owens Cross Roads AL, although it is listed as "no pets"
If you take a slightly longer route through Indiana:
Mastodon DGC in Fort Wayne is a fun, flat, mostly wooded round.
Other courses that I hope to play on that route:
Boylan Acres near Indianapolis IN
Patoka Lake, Niehaus Park, Prides Creek Park, and Mesker Park in southern Indiana.
I hope you have a great trip!