my @pids;
{
    open(my $pids_fh, '<', $pids_qfn)
        or die("Can't open PIDS file \"$pids_qfn\": $!\n");
    push @pids, /\|(\d+)$/ while <$pids_fh>;
}

my $pids_pat =
    map qr/$_/,
    join '|',
    #map quotemeta,  # We're only dealing with digits
    @pids;

open(my $fids_fh, '<', $fids_qfn)
    or die("Can't open FIDS file \"$fids_qfn\": $!\n");

while (<$fids_fh>) {
    print if /^$pids_pat;/;
}
[download]

If you need extra speed and your Perl is older than 5.10, change

my $pids_pat =
    map qr/$_/,
    join '|',
    #map quotemeta,  # We're only dealing with digits
    @pids;
[download]

to

use Regexp::List qw( );
my $pids_pat = Regexp::List->new()->list2re(@pids);
[download]

5.10 already does the optimisation Regexp::List does.