Include post title in readmore link?

If the following code is in-fact the code that is controlling your read more link then you perhaps the following may work;

if ($show_more_tag) {
    global $post;
    $excerpt = $excerpt . ' <a class="more-link" href="' . $perma_link . '">' . get_the_title($post->ID) . ' </a>';
}

I’ve declared global $post; again, just in case the above call to the conditional statement above is built outside of the initial $post shown in your first snippet; because you haven’t provided the full code to assess.

But try the above and see if it has any effect and let us know.