After you open LOG but before you print anything, add this code:

if (!(my $pid = open(STDOUT, "|-"))) {
    die "fork failed: $!" unless defined $pid;
    while (<STDIN>) {
        print $_;
        print LOG $_;
    }
    exit;
}
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This will fork a child process that will capture everything sent to STDOUT. The print statements appear to the child as STDIN. What you do with it is up to your code.