Bluff the police without lying. "Without looking at the fine print I believe I'm going to need to report this to BATFE. Can you give me specifics as to where the gun went missing? After I turned it in to Officer xxxxx where exactly did he place it and into whose custody did he consign it. Etc."
I wouldn't bluff, I'd pull out my phone and make the call, fill out the form when I got home. Watch them shit themselves as I'm a law-abiding citizen!