Mark Gritter suggested
here that the median badugi might be a bit better than imagined. This sounded logical, and not knowing how to approach the problem mathematically, I wrote a little engine that would deal a badugi hand. It would throw away pat badugis, but if it had a one-card badugi draw, it would pick the best draw (since it often had two) and take one off. When it hit, it recorded the results. The median badugi of this experiment was JT7, down from the QT that I had originally surmised.
caveat: my programming skills are a joke.