I don't think there's a way to change the User-Agent. But you can force SHOUTcast to serve the stream by adding a Semicolon ;
to the end of the URL. SHOUTcast will then ignore the User-Agent.
HTML5 example:
<!DOCTYPE html>
<html>
<body>
<audio controls>
<source src="http://72.13.82.82:5100/;"/>
<em>Sorry, your browser doesn't support HTML5 audio.</em>
</audio>
</body>
</html>
You can also verify the workaround by visiting http://72.13.82.82:5100/;
directly in browser. You should see some ICY-metadata followed by the mpeg stream.