Only 700 (even if long) rows? Then you don't need any disc-based hash. Just create a hash with the rs value (i.e. for example the 'rs3507865') of a row in file2 as key and the position in the file as data. The position in the file you can find with tell() (before reading the line).

Then just read the numbers in file1 and look up their position in the hash and use seek() in file2 to go there

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;
  }
}
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UPDATE: Added a '^' to the regex in the script