Filtered by a custom field, ordered by another

You are sorting it incorrectly. You will need to check for artist-status meta key with meta_query and sort by last-name metakey.

Here is your query.

$args = array(
    'post_type' => 'artist',
    'posts_per_page' => -1,
    'meta_key' => 'last-name',
    'orderby' => 'meta_value',
    'order' => 'ASC',
    'meta_query' => array (
        array(
            'key' => 'artist-status',
            'value' => 'invited',
            'compare' => '=',
        )
    )
);

$loop = new WP_Query( $args );