There is definitely not such a breed as a Miniature Beagle. Pocket beagles were bred in the 1300’s and 1400’s but there is no modern-day equivalent. In fact, the term has become synonymous with poor quality puppies bred for the pet market. Many breeds are represented by smaller or larger varieties which are in fact recognized as breeds in themselves (for example the Miniature Schnauzer and Miniature Poodle).

The American Kennel Club’s objective is to "Advance the study, breeding, exhibiting, running and maintenance of purebred dogs". I consider breeding from purebred dogs in order to achieve characteristics which are far from the AKC characteristics (or breed standard) quite irresponsible. Breed profiles exist so that experienced breeders can monitor the quality of their lines and select, based on this quality, which lines to continue in their breeding program.

As you so rightly suspected, breeding programs attempting to create animals so distant from the breed standard can produce animals with health or behavior problems, and generally lower life expectancies. There are many smaller dogs which look similar to beagles in shelters up and down the country which need homes; and who often have mixed parentage and ancestry.