If you can copy STDIN before anything reads from it, you might have luck. The following code works for me under Linux:

#!/usr/bin/perl -w

use strict;

open (NEWIN, "<&STDIN") or die "$!";
open (STDIN, "/dev/null");
print while (<NEWIN>);
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I saved it as 'dup.pl' and called it with cat dup.pl | ./dup.pl.

(Find out about <& at open.)

Of course, you can pass a reference to a typeglob containing the NEWIN fh to the CGI.pm constructor...