How to put “Read more” link in Custom Excerpt inside p tag?

One approach is:

//... 

// this is the resulted paragraph without the enclosing <p> and </p> 
$first_para_inner_text = $matches [1] [1]; // <-- the index changed

$link = get_permalink($post);

// rebuilding the p 
$first_para="<p>" . $first_para_inner_text .  ' <a href="'.$link.'">Read More</a></p>';

echo $first_para;

Not tested, but you may get the idea. Just need to modify/change some parts of your code with this.