If I understand well, you reach both the "http://www.foobar.com/appdir/chat/index.html" URL and the "http://www.foobar.com/lightstreamer/demos/ChatDemo/index.html" URL from the same apache running instance.

Hence, when you ask the http://www.foobar.com/appdir/chat/index.html page, it correctly finds the /lightstreamer/demos/ChatDemo/ls/lscommons.js resource.

There is no reason why it shouldn't find /lightstreamer/demos/ChatDemo/ls/lsengine.html resource.
Can you reach the page directly, either from the browser address bar or after putting a test "A" link on the chat demo page?
Can you find any refused request in your monitor or in your web server log?