By Evelyn Nieves The family—a couple and their four children, ages 5 to 11—wanted a dog in the worst way. Not just any dog, but the type more popular today than any of the dazzling breeds at the Westminster Kennel Club dog show. They wanted a labradoodle. With luck and money, they found one not far from where they live in Connecticut. The breeder claimed the dog came from several generations of labradoodles, who in turn were carefully bred from miniature poodles and Labrador retrievers in Australia, where labradoodles were popularized 25 years ago. A ball of chocolate fluff, the puppy cost $2,…