The docs for AnyEvent::HTTP (which I haven't used) show the headers being passed in to http_request() as a reference to a hash, not the hash itself. Try:

my $headers = {
    Content-Length => 2345,
};

http_request 'PUT'     => $url,
             'headers' => $headers,
              ...
[download]

I can't see a simple way to turn on tracing in AnyEvent::HTTP either, which isn't helping my debugging.

Maybe setting $ENV{ PERL_ANYEVENT_VERBOSE } see http://search.cpan.org/~mlehmann/AnyEvent-7.12/lib/AnyEvent.pm#ENVIRONMENT_VARIABLES