I think this is a bug in XML::DOM. A work-around is to print to a string and then output it yourself with the correct I/O layers in place:

  my $doc = $parser->parsefile("$srcdir/$_");
  my $output = $parser->toString;
  open(my $out, ">:utf8", "$dstdir/$_") or die "...";
  print $out $output;
  close($out);
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