After doing research into what makers these days are using to provide the brains for their robots, I have made a decision.
I read this great article from Make Magazine comparing the Raspberry Pi to the BeagleBone and the Arduino Uno. Contrary to their advice though, and despite how cute it would be to have a BeagleBone inside K-9, I ultimately chose the Raspberry Pi. Model 3 to be specific.
Their reasoning is that Arduino is lightweight and best for beginners and BeagleBone offers more pins for sensors and servos. However, I feel that the Rpi brings real computing power to the mix. Furthermore, the new Model 3 has built-in Wifi and Bluetooth LE. Since I'm planning for K-9 to be as autonomous as possible and not just a remote control toy, I feel that the Rpi is the only real choice.
The fact that today is (rounded) pi day (3.1416 = 03/14/16) just make it the sweetest pi(e) of all.