IMO, I would say that the customer is unknown for determining whether or not the reporting threshold ($25,000) has been met. If the threshold was met, I'd check the box for "suspect information unavailable" and detail everything you know in the narrative. I think that the most I would put in Part II is the name that you and law enforcement believe is the actual name, but even with that, you're not 100% certain.
Don't make me say, "I told you so!" Sincerely, your friendly Compliance Officer.