You can just append the read more button to the excerpt like below –
add_filter('lae_posts_carousel_entry_excerpt', 'mytheme_restrict_excerpt', 10, 1);
function mytheme_restrict_excerpt($excerpt) {
$excerpt = '<div class="entry-summary">';
$length = 120; // number of characters without breaking the words
$append = '...';
$string = get_the_excerpt();
$string = trim($string);
if(strlen($string) > $length) {
$string = wordwrap($string, $length);
$string = explode("\n", $string, 2);
$string = $string[0] . $append;
}
$excerpt .= $string;
$read_more_text = __('Read More', 'livemesh-el-addons');
$excerpt .= '<div class="lae-read-more">';
$excerpt .= '<a class="lae-button" href="' . get_the_permalink($post_id) . '">' . $read_more_text . '</a>';
$excerpt .= '</div><!-- .lae-read-more -->';
$excerpt .= '</div><!-- .entry-summary -->';
return $excerpt;
}