Best Barbeque near the Vandy campus is "Hog Heaven", next to Centennial Park. However, I realize just how sophisticated some of our HSBB websters might be -- especially those who might be visiting Vanderbilt. Hog Heaven looks terrible on the outside and the inside is worse -- but, if it's BBQ you want, Hog Heaven is the place.
As for Corky's. The Original Corky's in Memphis is excellent. Corky's in Nashville, in my opinion, not quite on par with the original. [My personal favorite BBQ in Memphis is Blues City Cafe on Beale Street.]
Now I've done it. I've set off a barbeque war on the internet. So barbeque me.