Dave_PA,
Here is how I would do it as a general solution which is a balance of reduced complexity, IO, and memory consumption (untested).

#!/usr/bin/perl
use strict;
use warnings;
use File::ReadBackwards;

my @input = qw/sun.txt mon.txt tue.txt wed.txt thu.txt fri.txt sat.txt
+/;

my %seen;
open(my $rev_out_fh, '>', 'output.rev') or die "Unable to open 'output
+.rev' for writing: $!";
for my $file (reverse @input) {
    my $bw = File::ReadBackwards->new($file) or die "can't read '$file
+' $!";
    while (defined(my $line) = $bw->readline)) {
        my $id = substr($line, 8, 5);
        next if $seen{$id}++;
        print $rev_out_fh $line;    
    }
}
open(my $out_fh, '>', 'output.txt') or die "Unable to open 'output.txt
+' for writing: $!";
my $bw = File::ReadBackwards->new('output.rev') or die "can't read 'ou
+tput.rev' $!";
while (defined(my $line) = $bw->readline)) {
    print $out_fh $line;
}
unlink 'output.rev'; # you may care if this fails
[download]

If everything fits into memory then this is probably unnecessary.