While Filter::Handle looks like the right module for the job, I'd probably just do it by hand as below. If your error output isn't going to be line-based, you should use select to poll for input within the child. Season to taste.

my $err_file = "error.log";
open(ERRLOG, "+>>logs/system/$err_file") or die "Could not open logfil
+e:$!\n";
flock(ERRLOG,2);

# Was *STDERR = *ERRLOG;
my $child = open(STDERR, '|-');
if ($child < 0) { die "Could not fork: $!\n" }
if ($child == 0) {
    while (<>) {
        printf ERRLOG '%s %s', scalar(localtime), $_;
    }
    exit 0;
}
print "One\n";
print STDERR "Foo\n";
print "Two\n";
print STDERR "Bar\n";

# and ends with:

flock(ERRLOG,8);
close(ERRLOG);
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