I wouldn't pay it.
It's not the "one dollar." It's the "having to tell them who I am" that would be the deal breaker. I don't see any way to give Facebook one dollar, or one penny for that matter, without giving them invaluable confirmation about who I really am.
That's not going to happen.
The question is... would Facebook actually terminate my account over the lack of one dollar? Remember - the company's biggest asset is the enormous number of people who use it. Terminating my account would hurt Facebook too - more than it would hurt me, imo.