I’ve seen a few suggestions around both Volokh and some other places that the first order of business is to challenge the NFA LEO sign off as arbitrary and capricious. Something to think about: first you really need to get the courts to recognize that an NFA item is an “arm” protected by the Second Amendment. If you don’t have a right to own something, than the government can get away with a lot more shennanigans than it can if you do. There might be grounds you could challenge LEO sign off on, but I’m not sure the Second Amendment is one of them, unless you can make a case that the arm in question is within its scope.
I agree with folks that the “common use” language presents a logical problem, but I don’t think it’s an insurmountable one. There are ways out of the trap, which I’ve talked about before, where if a law is interfering with determining whether an arm is “common” or “unusual and dangerous” one has to look at police use. I would also add that looking at commonality in countries which don’t have such restrictions on the item in question also needs to be considered.