LWP::UserAgent has its own methods of proxying which may work for you and is likely incompatible with Crypt::SSLeay proxy support.  
To use LWP::UserAgent proxy support, try something like:
 
my $ua = new LWP::UserAgent;
$ua->proxy([qw( https http )], "$proxy_ip:$proxy_port");

At the time of this writing, libwww v5.6 seems to proxy https requests fine with an Apache mod_proxy server.  It sends a line like:
 
GET https://www.nodeworks.com HTTP/1.1
to the proxy server, which is not the CONNECT request that some proxies would expect, so this may not work with other proxy servers than mod_proxy.  The CONNECT method is used by Crypt::SSLeayâs internal proxy support.