EAF, the Nat. Park would be an awesome place to play -- I think about it every time I'm around places like Elkmont, etc, which is often.
However, not to be a wet blanket, but I don't think there's a snowball's chance of you getting that permission. Understand, everyone in the park HATES the new weapons permitted rule. And there is absolutely no upside to them allowing an airsoft game. There's liability issues, safety issues, and PR issues. More power to you if you can get it to fly -- I'd be the first person to sign up for such a game -- but I think you'd have better odds of getting the president of Iran to host a Spring Break keg party in Panama City than you do of getting the Nat. Park Service to allow an airsoft game in the Smokies.