I know that there are a few Korean places off of University at least, but do we have anything like this (in Gainesville)?:
http://shilabbq.com/
I've been to a popular place in Orange County CA (Gen) on a few occasions, & the vibe is more upscale/modern - where your meals are self cooked at your table.
Seems like they'd absolutely kill it in JAX in a spot like Pele's.
It's not hip and could be better, but Sam Won on University does have grills at the booths.
Sam Won is the only place I know of. And last time I did grill it yourself was at least four years ago (always eat there often but let them cook in the kitchen)
You can buy everything you need to do it yourself. It's not that bad. I'm a fan of Gyu Kaku, a Japanese BBQ chain and almost the same as Korean BBQ