Ok here is a simpler solution (maybe not terribly elegant, but it works) - only for http, I hope SSL will be the same:

use strict;
use LWP::UserAgent;

my $localPort;
sub setLocalPort {$localPort = shift;}

package myHttp;

use vars qw(@ISA);
require LWP::Protocol::http;
@ISA = qw( LWP::Protocol::http );

sub socket_class {return "LWP::Protocol::http::Socket";}

sub _new_socket
{
  my($self, $host, $port, $timeout) = @_;
  my $s = $self->SUPER::_new_socket($host,$port,$timeout);
  main::setLocalPort($s->sockport());
  print "Local port: ",$s->sockport(),"\n";
  return $s;
}

package main;

LWP::Protocol::implementor( http => 'myHttp' );

my $ua = LWP::UserAgent->new(ssl_opts => { SSL_verify_mode => 'SSL_VER
+IFY_NONE'},);
$ua->agent('Mozilla/5.0 (compatible; MSIE 10.0; Windows NT 6.1; Triden
+t/5.0)');
$ua->max_size(50000000); # 50MB at most
$ua->timeout(45);

my $request = HTTP::Request->new(GET => "http://www.google.ch/");
$request->protocol('HTTP/1.0'); # we don't want chunked replies
$request->header('Accept' => '*/*');
$request->header('Accept-Encoding' => '');   # we don't want packed re
+sults
$request->header('Connection' => 'Close');
$request->header('Cache-Control' => 'no-cache');
my $resp = $ua->request($request);

if ($resp->is_success)
{
  my $data = $resp->content;
  print "$data\n";
  print "Local port: $localPort\n";
}
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