I tried the BodyFilter::save and it worked fine.
However, what I ended up using was HeaderFilter::simple.
It gave me enough information to show what I was missing.
Thanks for your help.

use warnings;
use strict;
use Data::Dumper;
use HTTP::Proxy;
use HTTP::Proxy::HeaderFilter::simple;
use IO::File;

my $proxy = HTTP::Proxy->new(port => 8081);
$proxy->host(undef);
my $filter = HTTP::Proxy::HeaderFilter::simple->new(\&myfilter);
$proxy->push_filter(request => $filter);
$proxy->start();

sub myfilter {
   my ($this, $headers, $message) = @_;
   print $headers->as_string();
   for my $pair (split(/\&/, $message->content())) {
      my ($key, $value) = split(/=/, $pair);
      $value = '' unless defined $value;
      printf "%s = %s\n", $key, $value;
   }
   STDOUT->flush();
}
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