Custom archive URL as a subfolder of the custom post type slug

You can have a different slug for the archive by setting has_archive to a string:

register_post_type(
    'whatever',
    array(
        'has_archive' => 'whatever/results',
        'rewrite'     => array(
            'slug' => 'whatever',
        ),
        // ...etc.
    )
);

I’ve tested that and it seems to do exactly what you needed.