I want to make him DNA test his dogs to prove what he's done. Whatever caused Joe's toothless wonders, the only thing DNA testing will prove is whether the parents of his dogs are or are not as listed on the papers. While it is theoretically possible that every single breed might have a unique DNA sequence somewhere in its genome, it will be a very long time before genetic researchers have the time or the money to pursue that kind of trivia. At present it is difficult to distinguish pure wolf from wolf-hybrid from dog. Some of this DNA stuff has to be wrong sometimes, too.
False readings are very unlikely.
Paternity tests use multiple markers to minimize the possibility of error or mutation skewing the result. The possibility that the test result would be incorrect are too minuscule to cause any serious concern. Human error, however, is another matter. As with any test, if the sample is mis-handled the results can be incorrect. Blood samples, due to the amount collected and method of collection, are least prone to contamination or mis-labeling.