Thanks for the pointer Kanji, I eventually figured out that my request needed a line like:
Authorization: Basic foobar

For the posterity of future monks, my final code looks like this:


snip

  my $userPass = base64("user:password");
    
    # Need two \n in the get request
    my $get =<<EOF;
GET $path HTTP/1.0
Authorization: Basic $userPass

EOF
    # Open the socket
    my $http_request = IO::Socket::INET->new(
                                             Proto    => "tcp", 
                                             PeerAddr => $host,
                                             PeerPort => $port,
                                             ) or return $!;

    print $http_request $get;  # Write the request

    my $webPage;
    { 
        # Grab the output
        local($/) = undef;
        $webPage = <$http_request>;
    }

    return $webPage;
[download]

snip