Ah, the games faders have to play. If you're going to do something like DOC says above, which might be your only play if you're really willing to potentially sign your life away to the cult, be sure to maybe speak with a lawyer about it. Maybe you could get something drawn up legally because otherwise they'll bring in their copies of your papers and they'll be sure to have people speak for you as to you being a good JW so they'll try to establish a pattern of you as a JW despite anything you do to circumvent it.
Or you could just establish a boundary and say that you won't do it, you won't give them one. Just say you're a private person or that you feel weird or something and be quirky and don't do what they say. I don't think they can disfellowship you because you don't give them a copy of your card. You will have to sack up and take a stand though. Or just let them bully you like they do with their bad cult boundaries and put your life in their hands. I'm pretty sure you've been given advice on this already.