In your first request, you're not sending the SOAPAction header. Also, most likely, if you need to speak SOAP, something like SOAP::Lite is better than constructing your SOAP requests manually.

In your second request, you are doing a very weird think when setting the headers via the ->default_headers method. Don't do that, but construct your request properly through HTTP::Request::Common instead.

I recommend doing a proper compare between what your browser sends and what your script sends, by using Wireshark and/or LWP::Debug.