I soldered the HC164 board:And tested it with simple code that simply sets the servo outputs to a fixed value. Here's the hardware:

The HC164 is powered with 3.3 V from the Teensy, which worked well.

Then I added a simple I²C slave interface and tested that with a Bus Pirate:

I used the Bus Pirate's on-board pull-ups with VPullup connected to the Teensy's 3.3 V output. It's been the first time I used that feature and it worked flawlessly.

The next step is to connect this setup to a Raspberry Pi and write code that sends servo positions over I²C. When that works, all I need is the Pi-Duino-Teensy board to put all of this together.