Since I don't have access to that site, I couldn't test at first, but I whipped up the following test server which I hope emulates the server you're accessing, and it seems that the solution provided by pmqs here works. The only issue I'm still having is that the last line is cut off, probably some interaction with WWW::Mechanize, I haven't checked it out yet. First the server (run e.g. via plackup):

use warnings;
use strict;
use IO::Compress::Gzip qw/$GzipError Z_PARTIAL_FLUSH/;

my $app = sub {
    my $env = shift;
    die "This app needs a server that supports psgi.streaming"
        unless $env->{'psgi.streaming'};
    die "The client did not send the 'Accept-Encoding: gzip' header"
        unless defined $env->{HTTP_ACCEPT_ENCODING}
            && $env->{HTTP_ACCEPT_ENCODING} =~ /\bgzip\b/;
    # Note some browsers don't correctly support gzip correctly,
    # see e.g. https://metacpan.org/pod/Plack::Middleware::Deflater
    # but we're not checking that here (and we don't set the Vary head
+er)
    return sub {
        my $respond = shift;
        my $zipped;
        my $z = IO::Compress::Gzip->new(\$zipped)
            or die "IO::Compress::Gzip: $GzipError";
        my $w = $respond->([ 200, [
                'Content-Type' => 'text/plain; charset=ascii',
                'Content-Encoding' => 'gzip',
            ] ]);
        for (1..10) {
            $z->print("Hello, it is ".gmtime." GMT\n");
            $z->flush(Z_PARTIAL_FLUSH);
            $w->write($zipped) if defined $zipped;
            $zipped = undef;
            sleep 1;
        }
        $z->print("Goodbye!\n");
        $z->close;
        $w->write($zipped) if defined $zipped;
        $w->close;
    };
};
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And the client:

use warnings;
use strict;
use Data::Dump;
use WWW::Mechanize;
use Compress::Zlib;

my $gunzip = inflateInit(WindowBits => 16 + MAX_WBITS)
    or die "Cannot create a inflation stream\n";

my $mech = WWW::Mechanize->new();
$mech->add_handler( response_data => sub {
    my ( $response, $ua, $h, $data ) = @_;
    my ($buffer, $status) = $gunzip->inflate($data);
    die "zlib error: $status" if length $status;
    dd $buffer;
    1;
});
$mech->get('http://localhost:5000');
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