Maybe the private areas should require a small fee be paid for access, that way you have a paypal trail. Pay Pal has contact with banks and the banks already have all sorts of unpleasant procedures to handle fraud with...
I know it wouldn't solve the greater problems, but what about an addition that someone must have been an active member for a year or so here, and have made enough posts to be eligible to apply for the private access? Similar to the system you have with moderation that is for starting to post in the public area. This is for private access. Sure, it would keep some people who might otherwise benefit from posting about things they don't want public from being able to do that for a while, but overall I'd imagine a lot less impact from something like this than if you required the whole scheme. Retina scan and blood sample for access next?
I mean, implement a system to actively test users credibility in advance to any private area access. One with enough time to observe and expose deceitful users. More time for all sorts of things that would catch the fakers. And in the end, its already you right to refuse anyone access. Just my opinion but try making it much harder to gain access first, and if problems remain then think about the other things.