Field can't be searched if it can't be edited or doesn't display. Setting it to display on profile, but place in a not shown on profile position satisfies the displayed check, but the edit check might be more complicated. Try using CB Conditional and applying a condition only to profile edit and no other view. Can also try using CB Privacy to set edit permissions to moderators only. Final option is try setting the field to read only, which will display it but can't be edited.