Make only the image an anchor link using the advanced excerpt plugin?

To get a link around only part of the excerpt content you will need a filter on get_the_excerpt similar to the following:

function dummy_excerpt_filter($excerpt) {
  // manipulate the excerpt
  return $excerpt;
}
add_filter('get_the_excerpt','dummy_excerpt_filter');

The problem with that is that the // manipulate the excerpt part means using regex on your excerpt content, which can be dicey. Before diving into that I am going to suggest another solution.

Instead of this…

<a href="https://wordpress.stackexchange.com/questions/104780/<?php the_permalink(); ?>">
    <?php the_excerpt(); ?>
</a>

Do this…

<a class="excerpt-link" href="https://wordpress.stackexchange.com/questions/104780/<?php the_permalink(); ?>">
    <?php the_excerpt(); ?>
</a>

You can now use the excerpt-link class to control the appearance specifically of the links in the excerpt. That may get you were you want to go without having to regex markup.

Since you seem to be editing the template anyway, you could also use:

$excerpt = get_the_excerpt();
// manipulate and echo the excerpt however you want.

Though that still may mean using regex on markup.