Could be an Apache bug, some googling for the error message showed an old bug ("mod_proxy fails to send post request to backend server"), usually blamed on IE. Another style of invocation does work:

my $req = HTTP::Request::StreamingUpload->new(
    PUT => "https://$server", 
    callback => sub { shift @chunk },
    headers => HTTP::Headers->new(
        'Content-Type' => 'application/binary',
        'Content-Length' => 9,
        'Accept' => 's',
    ),
);
my $res = $ua->request($req);
[download]

My httpd is 2.2.15 though (RH6), and it should have been fixed in 2.2.9