I would do something like the following ...

                                                                    
my %weight = ( );
foreach my $filename (keys %hoh) {
    my $psubhash = $hoh{$filename};
    foreach my $letter (keys %$psubhash) {
        my $weight    = $psubhash->{$letter}->{'weight'};
        my $intensity = $psubhash->{$letter}->{'intensity'};
        $weight{$weight} = [ $intensity, $filename ];
    }
}
                                                                      
+          
my @weight = sort { $a <=> $b } keys %weight;
foreach my $weight (@weight) {
    my $p = $weight{$weight};
    my ($intensity, $filename) = @$p;
    printf "weight: %4d intensity: %d Filename: %s\n",
        $weight, $intensity, $filename;
}
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Which gives:

weight: 1000 intensity: 4 Filename: FILENAME3
weight: 2000 intensity: 7 Filename: FILENAME2
weight: 3000 intensity: 2 Filename: FILENAME1
weight: 4000 intensity: 3 Filename: FILENAME1
weight: 5000 intensity: 3 Filename: FILENAME2
weight: 6000 intensity: 3 Filename: FILENAME3
[download]

Update:  Fixed reversed values for weight and intensity.