open(FILE2,...
my $position=tell(FILE2);
my %rs;
while ($line=<FILE2>) {
  my ($key)= $line=~/^(rs\d{5,})\b/;
  if (defined $key) {
    $rs{$key}= $position;
  }
  $position=tell(FILE2);
}

...
while (defined ($line= <FILE>)) {
...
foreach (@output) {
  if (exists $rs{$_}) {
    seek(FILE2,$rs{$_},0);
    my $line= <FILE2>;
    print FD $line;
  }
}