Yoast SEO breadcrumbs: how to create a filter that uses the url slug for breadcrumb titles

Yoast does have a filter for you to use. See here:
https://gist.github.com/jmcclellan/b2d97ffee0e924e28fa1

I used this to add “parent” pages to custom post types. We wanted to use pages as our category landers w/ custom post types as children. Yoast would not output the parent page slug by default since there is technically no real relationship, so we used this function to override and splice in the parent slug for each custom post type. You should be able to modify this solution for your needs:

add_filter("wpseo_breadcrumb_links", "wpse_100012_override_yoast_breadcrumb_trail");

function wpse_100012_override_yoast_breadcrumb_trail($links)
{
    $post_type = get_post_type();

    if ($post_type === 'claims') {
        $breadcrumb[] = array(
            'url' => '/claim',
            'text' => 'claim'
        );
        array_splice($links, 1, -3, $breadcrumb);
    }
    return $links;
}

For the lowercase styling, just use CSS:

#breadcrumbs {
    text-transform: lowercase;
}