One way to achieve this would be to use Selenium, and write a custom script to gather the output from it... But I'm sure that falls far beyond the scope of what you're attempting to do.
The way I would go would be to invert the responsibility. Have the JS send the output to a PHP endpoint, and use that output however you see fit.
Here's an example.
Javascript
<script>
var outputElement = 'html';
var HTML = $(outputElement).html();
var endpoint = 'myEndpoint.php';
$.post(endpoint, { html: HTML }, function(data) {
alert('Output sent');
});
</script>
One caveat here is that you will not get the DOCTYPE declaration, or any attributes on your HTML tag, if this isn't acceptable, you may reconstruct them in the PHP file below.
PHP
<?php
$html = $_POST['html']; // Be VERY CAREFUL with what you do with this...
// If you need to have the doctype and html tag... Use your own doctype.
// $html = sprintf('<DOCTYPE html><html class="my-class">%s</html>', $html);
// Do something with the HTML.
You have to be very careful when sending HTML over POST. If you're using this HTML to output on your website, it can easily be spoofed to reveal sensitive data on your website.
Reference