A hash of prioritised lists seems like what you need:

use warnings;
use strict;

my $inputdir;
my %files = Prioritise_requests ($inputdir);

for my $fileList (sort keys %files) {
    print "$files{$file}->[0]\n"; # Lowest priority file
    print "$files{$file}->[-1]\n"; # Top priority file
}

sub Prioritise_requests {
    my ($input_directory, $node) = @_;
    my %files;
    my $index = 1;
    
    while (1) {
        last if ! opendir SCAN, "$input_directory$index";
        
        for my $filePath (grep (/^Flat_file.*\.txt$/, readdir (SCAN)))
+ {
            my ($file) = $filePath =~ m![^\\/]*$!;
            push @{$files{$file}}, $_;
            ++$index;
        }

        closedir SCAN;
    }

    return %files;
}
[download]

Note that this is untested and doesn't include all the "non-issue" work of your sample code.

Updated to avoid "perpetual" loop