LWP tries to use Crypt::SSLeay when you request a page using HTTPS. Crypt::SSLeay doesn't use the proxy information that you have set up in LWP. You need to set the proxy information as an environment variable instead.

use strict;
use warnings;

use LWP::UserAgent;
use LWP::Debug qw(+);

$ENV{HTTP_Proxy} = 'http://192.168.2.1:8118';

my $browser = LWP::UserAgent->new(keep_alive => 1);
$browser->env_proxy(1);
my $url = "https://www.hsbc.co.uk/";
my $response = $browser->get($url);
print $response->as_string();
[download]

As a general point though - it looks as though you are trying to screen scrape a banking application. Take a look at WWW::Mechanize as a better way of managing navigation and form filling.