Here is a backtracking method which works on sorted data, dogs with rare letters first. (Similar to LanX's proposal.)

use strict;
use warnings;
use Data::Dumper;

sub score {
  my ($dog, $letters) = @_;
  my $sum = 0;
  $sum += $$letters{$_}//0 for grep {/\w/} split //, $dog;
  return $sum/length($dog);
}

my @n = qw(Affenpinscher AfghanHound AiredaleTerrier Akita AlaskanMala
+mute AmericanEnglishCoonhound AmericanEskimoDog AmericanFoxhound Base
+nji BassetHound Beagle BeardedCollie Beauceron BlackandTanCoonhound B
+loodhound BluetickCoonhound BorderCollie BorderTerrier Borzoi BostonT
+errier BouvierdesFlandres Boxer BoykinSpaniel Briard Brittany Brussel
+sGriffon BullTerrier Bulldog Bullmastiff ChesapeakeBayRetriever Chihu
+ahua ChineseCrested ChineseShar-Pei Chinook ChowChow ClumberSpaniel C
+ockerSpaniel Collie Curly-CoatedRetriever Dachshund Dalmatian DandieD
+inmontTerrier DobermanPinscher DoguedeBordeaux EnglishCockerSpaniel E
+nglishFoxhound EnglishSetter EnglishSpringerSpaniel GermanPinscher Ge
+rmanShepherdDog GermanShorthairedPointer GermanWirehairedPointer Gian
+tSchnauzer GlenofImaalTerrier GoldenRetriever GordonSetter GreatDane 
+Greyhound Harrier IbizanHound IcelandicSheepdog IrishSetter IrishTerr
+ier Kuvasz LabradorRetriever LakelandTerrier LhasaApso Lowchen Maltes
+e ManchesterTerrier Mastiff MiniatureBullTerrier MiniaturePinscher Mi
+niatureSchnauzer NeapolitanMastiff Newfoundland NorfolkTerrier Peking
+ese PembrokeWelshCorgi PharaohHound Plott PortugueseWaterDog Pug Puli
+ PyreneanShepherd RatTerrier ShibaInu ShihTzu SiberianHusky SilkyTerr
+ier SkyeTerrier SmoothFoxTerrier SoftCoatedWheatenTerrier SpinoneItal
+iano St.Bernard StaffordshireBullTerrier WireFoxTerrier WirehairedPoi
+ntingGriffon Xoloitzcuintli YorkshireTerrier);

my $letters = "A(8),B(2),C(5),D(8),E(7),F(1),G(3),H(9),I(8),K(1),L(9),
+M(1),N(12),O(13),P(1),Q(0),R(5),S(6),T(4),U(6),V(0),W(1),X(1),Y(0),Z(
+1)";
my %letters = map { /(\w)\((\d+)\)/; { $1 => $2 } } split /,/, $letter
+s;
@n = map { $_->[0] } sort { $a->[1] <=> $b->[1] } map { [ $_, score( $
+_, \%letters ) ] } @n;
$letters = join '', sort { length $a <=> length $b } map { /(\w)\((\d+
+)\)/; $2?$1 x $2:'' } split /,/, $letters;

sub dogtree {
  my ( $level, $tree, $letters, @dogs ) = @_;
  print "$tree\n" and exit unless $letters;
 DOG: while( my $dog = shift @dogs) {
    my $remaining = $letters;
    $remaining =~ s/$_//i or next DOG for grep { /\w/ } split //, $dog
+;
    dogtree( $level+1, "$tree $dog", $remaining, @dogs );
  }
}

dogtree 0, '', $letters, @n;
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Takes 14 seconds on my machine to find one solution.