You can't edit someone with same or higher permissions than you. Are you trying to edit someone with "usereditors" permissions while being a "usereditors" ? Does editing on profile work? Note CB Core Fields Ajax is an extension of a fields edit functionality so if the field is not marked to display in profile edit it won't do anything.

There's no functionality to assign users to other users like that. You'll end up making them full moderators which have moderation permission throughout the site. Only way to do that is probably a custom implementation with your own HTML, JS, and letting CB Auto Actions handle the save process. If you're ok with them being site wide moderators then you'd need the following usergroup setup.

- Registered
- - User Editors

This gives "User Editors" authority over Registered. Now make a view access level that includes "User Editors" and any other users you want with moderator permissions then select that view access level as the moderator view access level in CB > Configuration > Moderation.

Does it work from profile? Can they reach profile edit of that user from their profile? Make sure moderation is working first then we can worry about ajax editing, which has other requirements like the field being marked editable.