The problem is, if they get pissed at you for something, you give them all the ammo they need to potentially ruin your life, so....
The following applies to all afraid to pony up ID info...
That's the fear talking.
If they get pissed at you, they will blacklist you, but they don't want to ruin your life because it's bad for business and bad PR for anti-prostitution.
Most mongers behave for fear of blacklisting, to prevent a loss of what they love, which is more effective than punishment. If they punish you by ruining your life, you might decide to ruin their business which is a lot easier than your wife identifying you as a client from a large leaked monger database of burner#, partial ids if kept, and a list of providers seen, dates, etc.
LE if they get their hands on it won't publish it without getting sued. I also believe LSC destroys the ids after a couple of good sessions. They don't want this getting out accidentally, it will severly hurt their business. They don't need it to punish, when blacklisting is more effective, and less likely to cause trouble for them.
There are several levels of blacklist. You can be just blacklisted from the org. Things like if you walk without paying, complain too much, write bad reviews, etc. But they will let other orgs decide to qualify you. And, you can be contrite, pay for the lost payment, promise not to write a review if bad, etc.
However, if you are violent, rob, rape, any serious non-censual behavior, they will tell all the orgs about your bad behavior, blacklisting you everywhere.
"you give them all the ammo they need to potentially ruin your life"
So, potentially ruining your life is true if you get divorced from being outed to a wife or pursuing a political career. But you have already exposed yourself way more than any chance a Korg will out you.
Your wife doesn't need a korg to leak data, to figure out you are cheating. If you are doing bbfs, the risk of you giving her an STD is much higher than you getting outed by the org.
The other constructive suggestion is to figure out how to deal with absences away from work, having a plausible story. The perfect cover would be if you work as a police detective or for the FBI. "Honey, I can't tell you where I've been, it's investgation policy.". You can even scan and remove tracking devices on your car under the policy.
If you are worried about ID theft from giving them a partial ID and photo, then why don't you have a fraud alert at all 3 credit agencies? This is way more effective and cheaper than subscrption to credit monitoring agencies.