After some experimenting and digging through various modules, I couldn't make the HTTP::Cookies thing work either. However, the following code sends the cookie (according to a trace from ethereal), so should hopefully do what you want:-

#! perl

use strict;
use warnings;

use SOAP::Lite;
use HTTP::Headers;

my $hdr = HTTP::Headers->new(
    Cookie => "session=my_session_id",
);

print SOAP::Lite
      -> uri('http://www.soaplite.com/Temperatures')
      -> proxy('http://services.soaplite.com/temper.cgi',
               default_headers => $hdr,
         )
      -> f2c(100)
      -> result;
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