We did spend a couple days at the club and I found out why they required a picture of us in advance. Luann found and booked the cottage thru an Airbnb. It was the property owner that requested the picture so the Club asked for it. The Cottage actually is on the property owners place at the club. He was a super guy, he couldn't have been anymore helpful and friendly. We could tell he is a very upstanding member in that club.
Clubs of course would want to bar true sex offenders. But my understanding is that some states require people convicted of public indecency, which can include simple nudity, or even (in some jurisdictions) public female toplessness, to register as sex offenders. It would be unfortunate if a search on a free national public sex offender's list might lead clubs to bar some people whose only crime was practising nudism at the wrong time and place. I wonder if any such sex offender list provides information on the underlying charges so clubs could differentiate between people listed due to minor versus major infractions.