Have reviewed your install and found the cause. It's "Allow Moderators to Edit User Profiles" within CB > Configuration > Moderation that's doing it. This is checked for fieldclass API calls, but should only be checked for frontend calls and not backend calls. I've set it to "Super Administrators Only" instead of "No" to resolve your issue. Note this change only allows Super Users to edit profiles so there should be no cause for concern leaving it as that for the time being. Have added a bug ticket to fix the fieldclass backend access check.
forge.joomlapolis.com/issues/6796