I think you need to give FastCGI its own filehandles to play with:

my $out = new IO::Handle;
my $err = new IO::Handle;

# keep the environment a global because of CGI.pm
my $req = FCGI::Request(\*STDIN, $out, $err, \%ENV);

while ($req->Accept() >= 0) {

   # generate the page

   select $out;
   print $page;
   select STDOUT;
}
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