Instead of trying to find out what you are missing, it is a lot easier (and faster) to just exclude what you have.

sub find_gaps {
  my $dates = shift;
  my @dates = sort{ $a->[0] <=> $b->[0] || $a->[1] <=> $b->[1] }
              map{[substr($_, 0, 4), substr($_, 5)]} keys %$dates;
  my( $minY,$minM,$maxY,$maxM ) = (@{$dates[0]}, @{$dates[-1]});
  my @gaps;
  for my $y ($minY .. $maxY) {
      exists $dates->{"$y-$_"} ? next : push @gaps, "$y-$_"
        for ($minY == $y ? $minM : 1) .. ($maxY == $y ? $maxM : 12);
  }
  return \@gaps;
}
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This code sorts to finds the min and max of the date range, then loops over the entire range in order looking to see if we have that date already. If we dont have the date then keep it to return.