FWIW if the directory structure is always DIR$N/001.$EXT, it has a fixed depth and you don't even need File::Find(::Rule), a simple glob is enough:

use strict;
use warnings;
my %seen;

while (my $file = glob 'Dir*/*') {
     (my $base = $file) =~ s/\.\w+$//;
     next if $seen{$base};
     # work with $file here
}
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