this primitive branch-and-bound will eventually spit out a solution but it might take days.

use strict;
use warnings;

use Data::Dump qw/pp/;
use feature qw/say/;


my %max =(
      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 @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);

#%max=( A=>1,D=>2,F=>1,I=>1,O=>1,R=>1,U=>1);
#@n = qw/ HONDA TOYOTA AUDI FORD/;


my @path;
our $level=0;

sub rec {
  my ($idx0,%max) = @_;
  #pp $idx0,\%count;
  local $level=$level+1;
  
 WORD:
  for (my $idx = $idx0; $idx <@n; $idx++){
    #say "*** ",
    my $word = $n[$idx];
    my %count=%max;
#    say " " x $level ,$word;
    for my $char ( grep {/[A-Z]/} split //, uc($word) ) {
      next WORD if $count{$char}-- <=0;
    }

    push @path, $word;
    unless (grep { $_ != 0 } values %count ) {
      print "RESULT: ", join ",",@path,"\n";
#      return;
      goto STOP;
    }
    
    rec($idx+1,%count);
    pop @path;
  }
}

rec(0,%max);
STOP:
[download]

a smarter approach would be to count the letters for each word in an initialization phase and to sort them according to exotic letters.

for instance if Z is 0 there is no point to check words with a Z in the following branch.

So start to find all allowed combinations for the most rare letter and continuing with the next rare one should greatly improve runtime.

Unfortunately I have no time to fiddle with entertainment problems, I hope my input helps! =)