Is there any way people could appear to be complying with the borganization's edict to sign it, but really not? Like signing it the wrong way, or having a trusted friend sign your signature and be a witness to the fact that you did not sign it? And have a letter on file in your home stating you did not sign it and were only presenting the pretend document to get the cult off your back? Sign it "Scooby Doo"?
I personally would be very worried that somehow the no blood card would get enforced.