He actually does train BJJ or at least with people versed in BJJ. He trains with a guy I know pretty well named Andrew Peterson. We're from the same area and I rolled with him a few years ago. I don't know how frequently Curran actually trains because he also does theater and is pursuing acting. But yeah, Andrew Peterson is one of the guys who trains with him, probably his main local grappling coach and he is a BJJ brownbelt. I'm sure they train exclusively no-gi though, so you might just call it grappling.
Curran got his own gym when he prepares for his catch wrestling matches. When he prepares for MMA he trains at Murcielago MMA.
I'll add that Curran did compete in Naga and he does also compete in MMA.
He competed once in Naga like 5 years ago. He also have 1 pro fight.