Who told you that LWP does not work with https?

from the LWP docs:
HTTPS Requests
HTTPS requests are HTTP requests over an encrypted network connection using the SSL protocol developed by Netscape. Everything about HTTP requests above also apply to HTTPS requests. In addition the library will add the headers ``Client-SSL-Cipher'', ``Client-SSL-Cert-Subject'' and ``Client-SSL-Cert-Issuer'' to the response. These headers denote the encryption method used and the name of the server owner.
The request can contain the header ``If-SSL-Cert-Subject'' in order to make the request conditional on the content of the server certificate. If the certificate subject does not match, no request is sent to the server and an internally generated error response is returned. The value of the ``If-SSL-Cert-Subject'' header is interpreted as a Perl regular expression.