If I recall correctly, the World Championship event in NYC has a similar rule, nail heads must be visible.
I can't confirm because it seems all event info is gone from the GNYC Council website.
But all it takes is a little hole drilled in the body to expose the tip of the nail.
For districts last year my son ran the same car he ran at the Pack. He built a completely new car for Worlds, but we tested that car against his Pack/District car and it was actually slower, so he went back to the Pack/District car for the World Championships.
I think it all depends on your District. If there is any way you can find out if there are typically a lot of cars entered in Districts, maybe you want to give your scout a better chance and work on the car more. If the car did great at the Pack races and he's up against just a few other cars at District, it might be better to just run with the car as-is and not risk messing up a good thing.