The buttons were moved to an external input device (currently breadboarded), which now sits next to the main breadboard:
The Teensy 3.1's inputs are 5V tolerant, so I can safely connect the Teensy 2.0's Tx pin to the Teensy 3.1's Rx pin without additional hardware. However, I should add some protection before things blow up anyway. Fortunately, I still have a spare 3.1 somewhere.
The two devices are still powered individually until I finally decide which side will have the battery.