Set the publication date independent of the scheduled publication date

I’m afraid it won’t be possible without any coding, but… It shouldn’t be so hard to achieve.

I would approach it this way:

1. Schedule post for publication using native post date.
2. Store the real publication date as some custom field.
3. Add custom action to transition_post_status hook and replace the publish date with the one stored in custom field.

.

function replace_publish_date( $new_status, $old_status, $post ) {
    if ( 'publish' == $new_status && 'future' == $old_status ) {
        wp_update_post( array(
            'ID' => $post->ID, 
            'post_date' => get_post_meta( $post->ID, 'real_publish_date', true )
        ) );
    }
}
add_action( 'transition_post_status', 'replace_publish_date', 10, 3 );