As we get more and better and faster tech it should be very easy to conduct a background check anywhere without much outlay for the equipment. It can't involve needless red tape, especially considering that from a realistic perspective damn near everyone at a gun show either already owns a gun or came along with a friend to see them and can legally buy one. The average gun show is more like a fan convention then a retail store and to my knowledge they're not where criminals typically go to purchase their guns. You could probably get the job done by treating the show like a bar where you show ID at the door and they run a quick check to see if you can legally buy a gun, and you get a plastic bracelet.

The problem though remains that a gun is not a carton of smokes or a six pack, IDing people for these is pretty effective because while a person can find someone to buy beer for them illegally and hand it off, it is hard for them to get someone to do this often enough to support a daily habit. You only need to buy a gun one time, and it isn't that hard for a felon to find a friend or bribable person to buy a gun for them. That's make it not quite a pointless gesture but not too effective either, so I'm okay with it if it represents a minimal cost and hassle but not otherwise.