You clearly did not try the code given by InfiniteSilence against the HTML returned by LWP, because if you had you would see that it worked perfectly. It is running against the "Javascript version" of the page.

By the way the answer to your original question is that no, it is not possible to do that without browser automation, because you need not only a Javascript interpreter but a full-fledged DOM engine and HTML processor of exactly the sort contained in web browsers, in order to make sense of that script.