Media uploader & manager: add/remove class on image with checkbox

Given this

I can’t use post meta

I suppose this is a typo (my emphasis)

setting a global meta key/value on the image is not OK

It’s either a post meta or a site option.
To use it as site option, use this in attachments_fields_to_edit:

$is_profile_image = get_option( '_is_profile_image' );
$checked = isset( $is_profile_image[ $post->ID ] ) ? 'checked' : '';

And this on attachment_fields_to_save:

function my_save_profile_image($post, $attachment) 
{
    $is_profile_image = get_option( '_is_profile_image' );
    if( isset( $attachment['is_profile_image'] ) )
        $is_profile_image[ $post['ID'] ] = true;
    else 
        unset( $is_profile_image[ $post['ID'] ] );

    update_option( '_is_profile_image', $is_profile_image );
    return $post;
}

The site option _is_profile_image will be an array like:

array (size=2)
  96 => boolean true
  97 => boolean true