This seems to work ok? (Note the extra line!):

#!/usr/bin/perl --

use strict;
use warnings;

use WWW::Mechanize 1.54;

my $ua = WWW::Mechanize->new;
$ua->add_handler(
    "request_send",  sub {
        $_[0]->content('');
        $_[0]->dump;
        return
    }
);

$ua->add_handler(
    "response_done", sub {
        $_[0]->content('');
        $_[0]->dump;
        return
    }
);

my $uri = URI->new('http://jigsaw.w3.org/HTTP/Basic/');
$ua->credentials( $uri->host_port, qw' test guest guest ');

$ua->get( $uri );

print "\n--- SPECIFY HOST MANUALLY ---\n";

my $host = $uri->host( $ua->res->header('Client-peer') );
$ua->credentials( $uri->host_port, qw' test guest guest '); ### ADDED 
+##
$ua->get( $uri, Host => $host );

__END__
GET http://jigsaw.w3.org/HTTP/Basic/
Accept-Encoding: gzip
User-Agent: WWW-Mechanize/1.66
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