Best I could do so far, strict and warning clean, at 158 chars:

sub path {
  my($g,$s,$e,@p,$q)=@_;@_=[0,$s];{my($l,@r)=@{+shift};if(!@p||$q>$l){
+@p=@r,$q=$l if$r[-1]eq$e;my$o=$$g{$r[-1]};push@_,map[$l+$$o{$_},@r,$_
+],keys%$o}@_&&redo}@p
}
[download]

   MeowChow                                   
               s aamecha.s a..a\u$&owag.print