I now need a sport bike for the other part of my riding. It has to be at the exact opposite spectrum. Any thoughts?
Each of us has his own spectrum, so it's a little hard to recommend without knowing yours. What kind of roads do you like? How fast are the turns? How important is power? How far will you ride? One-up or two-up? Is comfort a factor? Any need to carry more stuff than a tank bag can accommodate?
As an "anti-K12", I might look for a lightweight twin sans plastic. Suzuki's SV650 is incredibly entertaining and, in the right hands, will leave a big sport bike in the dust on a tight road. And though I'm not really a Ducati kinda guy, the Monsters appeal to me.
You also might consider bikes like the Triumph Speed Triple, Honda 919, and Yamaha FZ1. All are much lighter than the K12 but with comparable power.
Since you're happy with your BMW, you might consider one of their twins, such as the R1100S or R1150R. With either, you get reasonable passenger accommodations and, if you need carrying capacity, available factory luggage.